f 


PROBLEMS 


IN 


DESCRIPTIVE  GEOMETRY 


FOR  CLASS  AND  DRAWING  ROOM 


A  COLLECTION  OF  OVER  900  DEFINITE  PROBLEMS, 

FOR  STUDENTS   IN   ENGINEERING  AND   TECHNICAL  SCHOOLS. 

GENERAL  PROBLExMS,  SPECIAL  CASES,  APPLICATIONS, 

WITH  85   PRACTICAL  FIGURES. 


BY 


WALTER  TURNER  FISHLEIGH,  A.B.,  B.S. 

Department  of  Mechanism  and  Drawing 

University  of  Michigan 


OF   THE 

UNIVERSITY 

OF 


PUBLISHED 

BY   THE   A  UTHOR 

ANN  ARBOR,  MICH. 


Copyright,  igio 
By  W.  T.  fish  LEIGH 


The   Ann    Arbor    Press 


PREFACE 

GREAT  changes  have  been  made  in  recent  years  in  the  courses  offered  in 
Technical  and  Engineering  Schools,  but  in  none  has  the  advance  been 
more  marked  than  in  Descriptive  Geometry.  Engineers  and  instructors 
early  came  to  appreciate  the  practical  value  of  a  thorough  course  in  Descriptive 
Geometry  and  its  applications,  and  at  the  same  time  to  realize  that  a  working 
knowledge  in  such  a  subject  could  not  be  ^iven  to  the  student  without  dozens,  if 
not  to  say  hundreds,  of  original  problems.  The  theory  is  comparatively  simple, 
the  figures  in  the  text  easily  comprehended,  but  for  the  original  lay-out,  the 
special  case,  and  the  practical  application,  the  student  must  be  prepared  by 
constant  problem  practice.  Thus,  just  as  the  modern  Law  School  has  in  many 
instances  been  led  to  adopt  the  "case  method"  of  instruction,  so  has  the  Engineer- 
ing School  advanced  to  a  method  of  instruction  which  might  well  be  termed  the 
"Problem  Method." 

It  is  to  meet  the  deinand  for  a  complete  set  of  original  problems  in 
Descriptive  Geometry,  which  might  be  had  by  the  student  at  a  low  price,  that  this 
book  has  been  arranged.  The  aim  of  the  author  has  been  two- fold :  first,  to  present 
to  the  student  for  reference  and  study,  a  collection  of  definite  problems  under 
each  of  the  common  general  principles,  including  special  cases  and  groups  of 
applications  illustrating  the  use  of  these  principles  in  practical  problems ;  second, 
to  furnish  for  the  instructor  a  book  of  simple  notation  in  which  he  can  readily 
find  the  particular  sort  of  a  problem  desired,  and  of  such  scope  and  arrangement 
as  to  be  convenient  for  use  in  assigning  problems  for  blackboard  work,  for  home 
study,  or  for  drawing-room  solution.  In  attempting  to  produce  a  thoroughly 
usable  book,  special  attention  has  been  paid  to  the  following: — 

1.  Lay-outs.  Adopting  a  12"  X  18"  sheet  as  standard,  problems  have  been 
carefully  arranged  for  ^,  ^,  3^,  or  1  whole  sheet,  as  the  case  may  be.  A 
number  in  brackets  following  the  regular  problem  number  indicates  at  once  what 
part  of  a  sheet  is  needed  for  the  solution  of  each  problem,  so  that  several  prob- 
lems may  be  arranged  conveniently  on  a  large  sheet,  or  each  worked  upon  a 
separate  sheet  of  the  necessary  size.  For  class  blackboard  work,  the  student  has 
only  to  select  some  scale  convenient  for  the  problem  at  hand. 

2.  Notation.  Points  are  located  by  co-ordinates  which  refer  to  the  three 
common  co-ordinate  planes,  H,  Y  and  P.  A  right  line  is  located  by  giving  the 
co-ordinates  of  two  of  its  points.  A  plane  is  determined  by  locating  the  point 
of  intersection  of  its  H  and  V  traces  with  the  ground  line,  then  giving  the  angles 
which  its  traces  make  respectively  with  the  ground  line. 

This  system  has  been  tried  in  classes  at  the  University  of  Michigan  with  striking 
success.  The  notation,  being  extremely  simple  and  in  strict  accord  with  the  notation  ot 
Solid  Analytic  Geometry  and  Trigonometry,  is  mastered  by  the  student  in  one  lesson.  In  the 
use  of  these  co-ordinates  for  a  given  lay-out,  the  student  must  "think  in  space"  and  get  in 
mind  at  the  very  outset  the  general  location  of  points,  lines  and  planes  with  respect  to  one 
another  and  the  planes  of  projection.  He  is  aided  in  "seeing"  his  problem  and  understanding 
"what  he  is  about"  by  the  very  notation  by  which  he  locates  given  projections  or  traces  upon 
the  drawing  sheet.  For  the  instructor,  a  problem  is  easily  "read"  and  at  a  glance  he  can 
select  such  a  lay-out  as  he  may  desire. 

3.  Quadrants.  Problems  are  given  in  all  four  quadrants,  with  a  majority, 
however,  in  the  third  and  first  in  order  to  familiarize  the  student  with  the  arrange- 
ment of  views  for  these  two  quadrants,  which  are  commonly  used  in  practice. 

4.  Practical  applications.  A  large  number  of  applications  have  been  given 
to  interest  the  student  in  his  work,  to  give  him  at  first  hand  a  knowledge  of 


294669 


iv  preijpace; 

the  way  the  different  problems  come  up  in  practice,  and  to  give  him  as  soon  as 
possible  a  working  knowledge  of  Descriptive  Geometry  through  problems  where 
ingenuity  and  originality  in  applying  the  principles  are  as  essential  as  the  general 
analyses  and  figures  of  the  text.  The  trouble  with  ordinary  engineering  students 
seems  to  be,  not  so  much  that  they  do  not  knozt'  their  Descriptive  Geometry,  as 
that  they  cannot  apply  it.  In  reply  to  the  possible  criticism  that  a  few  of  the 
problems  offered  are  somewhat  more  artificial  than  practical,  the  author  has  only 
to  say  that  they  illustrate  principles,  that  they  interest  the  student  more  than  mere 
point-line-plane  problems,  and  that  pedagogically  they  have  been  found  by  experi- 
ence to  be  of  value. 

5.  Figures.  A  large  number  of  practical  figures  are  presented  on  blue  prints 
not  only  for  use  in  the  particular  problems  in  this  book,  but  with  the  hope  of  giving 
to  the  student  and  instructor  a  suggestion  of  the  many  useful  applications  of 
the  subject  in  hand,  and  thus  adding  to  his  study  an  interest  which  perhaps 
could  be  stimulated  in  no  other  way.  It  had  been  the  intention  of  the  author  to 
have  plates  made,  but  at  the  suggestion  of  practically  all  the  instructors  who  have 
used  the  book,  the  figures  are  retained  in  blue  print  form.  They  give  to  the 
student  at  first  hand,  excellent  examples  of  the  kind  of  draughting  which  should 
be  aimed  at  in  a  drawing  course,  and  make  valuable  reference  sheets,  as  the  work 
progresses. 

6.  Arrangement  and  grouping.  General  grouping  is  made  under  (1)  pre- 
liminary problems  on  the  point,  line  and  plane,  (2)  general  problems  based 
thereon,  (3)  representation  of  surfaces,  (4)  planes  tangent  to  surfaces,  (5)  inter- 
sections and  developments.  Under  each  of  these  general  groups  are  arranged 
smaller  groups  which  cover  general  cases,  special  cases  or  variations,  and  practical 
applications.  Particular  attention  has  been  paid  to  arrangement  and  headings, 
with  a  view  to  making  the  various  problems  easy  of  access  and  the  book  as  a 
whole,  convenient  for  the  assigning  of  problems  with  desirable  variation  and 
sequence. 

7.  Scope.  The  scope  of  the  book  is  thought  to  be  sufficient  for  any  element- 
ary course  in  Descriptive  Geometry. 

The  author  would  appreciate  any  suggestion  as  to  additions  or  changes  which 
it  might  seem  desirable  to  make.  While  a  large  number  of  the  problems  included 
are  original,  he  has  felt  free  in  consulting  other  works,  and  would  mention  the 
following  to  which  he  is  indebted  for  many  valuable  ideas  and  suggestions : 

ELEMENTS  OF  DESCRIPTIVE  GEOMETRY A.   E.   CHURCH. 

DESCRIPTIVE  GEOMETRY J.  A.    MOVER. 

ELEMENTS  OF  DESCRIPTIVE  GEOMETRY C.  W.   MAC  CORD. 

NOTES  ON  DESCRIPTIVE  GEOMETRY J.  D.  PHILLIPS  AND  A.  V.  MILLAR. 

PROBLEMS  IN  DESCRIPTIVE  GEOMETRY G.   M.  BARTLETT. 

CARPENTRY  AND  SHEET  METAL  WORK.  .  .AMERICAN  SCHOOL  OF  CORRESPONDENCE. 
DESCRIPTIVE  GEOMETRY  FILES UNIVERSITY  OF  MICHIGAN. 

The  author  wishes  also  to  acknowledge  the  kind  assistance  and  continued  encour- 
agement of  his  colleagues,  Messrs.  F.  R.  Finch  and  D.  E.  Foster  of  the  Univer- 
sity of  Michigan. 

W.  T.   FiSHLElGH. 

Ann  Arbor,  Mich.,  June,  1910. 


CONTENTS 

INTRODUCTORY    

PRELIMINARY  PROBLEMS. 


TO  FIND  THE  H  AND  V  PROJECTIONS  OF  POINTS 3 

TO  FIND  THE  H  AND  V  PROJECTIONS  OF  LINES 3 

TO  DETERMINE  THE  TRACES  OF  PEANES 5 

TO  ASSUME  RIGHT  LINES  IN  GIVEN  PLANES 5 

REVOLUTION  OF  POINTS  AND  LINES  ABOUT  GIVEN  AXES 5 

PROFILE   PROJECTIONS  OF   POINTS G 

PROFILE  PROJECTIONS  OF   LINES G 

PROFILE    TRACES    OF    PLANES G 

PROBLEMS  ON  THE  POINT,  LINE  AND  PLANE. 

L     TO   FIND  THE   POINTS   IN   WHICH   A   GIVEN    RIGHT   LINE 

PIERCES  THE  PLANES  OF  PROJECTION 7 

TO  FIND  POINTS  WHERE  LINES  PIERCE  PLANES  PARALLEL  TJ  THE  PLANES 
OF    PROJECTION     7 

TO  FIND  THE  PROJECTIONS  OF  LINES  WHICH  PIERCE  THE  PLANES  OF  PRO- 
JECTION   AT    GIVEN    POINTS 7 

APPLICATIONS    8 

2.  TO  FIND  THE  TRUE  LENGTH  OF  A  RIGHT  LINE  JOINING 

TWO  GIVEN  POINTS  IN  SPACE 8 

TO  FIND  A  POINT  ON  A  GIVEN   LINE  A  GIVEN  DISTANCE  FROM  ONE  END 9 

TO  FIND  THE  ANGLE  A  GIVEN   RIGHT  LINE  MAKES   WITH    H   AND  V 9 

TO  FIND  THE   PROJECTIONS  OF  A   RIGHT   LINE    MAKING  GIVEN   ANGLES   WITH 

H    AND    V 9 

TO  FIND  THE  THREE  PROJECTIONS  OF  A  POINT  ON  A  GIVEN  LINE,  EQUIDISTANT 

FROM   BOTH    H   AND  V 9 

APPLICATIONS 9 

3.  TO  PASS  A  PLANE  THROUGH  THREE  GIVEN  POINTS 10 

TO  PASS  A  PLANE  THROUGH  TWO  INTERSECTING  LINES 10 

TO  PASS  A   PLANE  THROUGH    TWO   PARALLEL   LINES LI 

TO  PASS  A  PLANE  THROUGH  A  POINT  AND  A  RIGHT  LINE 11 

APPLICATIONS     11 

4.  TO  FIND  THE  TRUE  SIZE  OF  THE  ANGLE  BETWEEN  TWO 

GIVEN  INTERSECTING  LINES  AND  TO  FIND  THE  PROJEC- 
TIONS OF  ITS  BISECTOR 12 

APPLICATIONS ,    12 

5.  TO  ASSUME  CERTAIN  LINES  IN  GIVEN  PLANES 13 

TO  LOCATE  CERTAIN   FIGURES  IN   GIVEN   PLANES 14 

APPLICATIONS     14 

6.  TO   FIND   THE   LINE   OF   INTERSECTION    OF   TWO    GIVEN 

PLANES    15 

APPLICATIONS      , 15 


Vi  CONTENTS 

7.    TO   FIND   THE   POINT   IN   WHICH   A   GIVEN   RIGHT   LINE 

PIERCES  A  GIVEN  PLANE 16 

WHEN  PLANE  IS  GIVEN  P.Y  TWO  RIGHT  EINES  IN   IT,  TO  BE  SOEVED  WITHOUT 

FINDING    TRACES     16 

APPEICATlONS    17 

8  THROUGH  A  GIVEN  POINT,  TO  DRAW  A  LINE  PERPENDIC- 
ULAR TO  A  GIVEN  PLANE,  AND  TO  FIND  THE  DISTANCE 
FROM  THE  POINT  TO  THE  PLANE 18 

THROUGH  A  GIVEN  POINT,  TO  PASS  A  PEANE  PERPENDICULAR  TO  A  GIVEN 
PLAN  E     18 

THROUGH  A  GIVEN  LINE  IN  A  GIVEN  PLANE,  TO  CONSTRUCT  A  PLANE  PERPEN- 
DICULAR TO  THE  GIVEN  PLANE 18 

APPLICATIONS    18 

9.  TO  PROJECT  A  GIVEN  RIGHT  LINE  UPON  A  GIVEN  PLANE. .   19 

TO  PROJECT  GIVEN  FIGURES  UPON  GIVEN  PLANES 19 

APPLICATIONS 19 

10.  THROUGH  A  GIVEN  POINT,  TO  PASS  A  PLANE  PERPENDIC- 

ULAR TO  A  GIVEN  RIGHT  LINE 20 

THROUGH    A    GIVEN    POINT,    TO    CONSTRUCT    A    LINE    PERPENDICULAR    TO    A 

GIVEN    LINE    20 

APPLICATIONS    80 

11.  TO  PASS  A  PLANE  THROUGH  A  GIVEN  POINT  PARALLEL  TO 

TWO  GIVEN  RIGHT  LINES 21 

TO  PASS  A  PLANE  THROUGH  ONE  LINE  PARALLEL  TO  ANOTHER 21 

TO  PASS  A  PLANE  THROUGH  A  GIVEN  POINT  PARALLEL  TO  A  GIVEN  PLANE  AND 

FIND  DISTANCE  BETWEEN   THE  TWO   PLANES 21 

APPLICATIONS    21 

12.  TO  FIND  THE  DISTANCE  FROM  A  GIVEN  POINT  TO  A  GIVEN 

RIGHT  LINE 22 

TO  FIND  THE  PROJECTIONS  OF  A  LINE  THROUGH  A  GIVEN  POINT  PERPENDIC- 
ULAR TO  A  GIVEN  RIGHT  LINE 22 

TO  FIND  THE  DISTANCE  BETWEEN  TWO  PARALLEL  LINES 22 

APPLICATIONS    22 

13.  TO  FIND  THE  ANGLE  WHICH  A  GIVEN  RIGHT  LINE  MAKES 

WITH  A  GIVEN  PLANE  23 

APPLICATIONS    23 

14.  TO  FIND  THE  ANGLE  BETWEEN  TWO  GIVEN  PLANES 24 

ONE  TRACE  OF  A  PLANE  BEING  GIVEN,  AND  THE  ANGLE  WHICH  THIS  PLANE 

MAKES  WITH  A  PLANE  OF  PROJECTION,  TO  FIND  THE  OTHER  TRACE 25 

APPLICATIONS    25 

15.  TO  FIND  THE  SHORTEST  DISTANCE  BETWEEN  TWO  RIGHT 

LINES  NOT  IN  THE  SAME  PLANE 26 

APPLICATIONS    26 

GENERAL  PROBLEMS   BASED   ON   POINT,  LINE   AND  PLANE, 

WITH  APPLICATIONS  27-32 

BUILDING  UP  SOLIDS,  CONDITIONS  GIVEN. 

PYRAMIDS,  CONES,  PRISMS,   CUBES 33-34 


CONTENTS  vii 

REPRESENTATION  OF  SURFACES 
HELICAL  CONVOLUTES.    assume  i;le;ments;  intersection  with  h  or 

OBEIQUE  PLANE  ;  ASSUME  POINTS  ON  SURFACE 35 

HYPERBOLIC   PARABOLOIDS  AND  CONOIDS,     assume  elements, 

FIRST    AND    SECOND    GENERATIONS;    PI,ANE    DIRECTERS    OF    BOTH    GENERA- 
TIONS ;  ASSUME   POINTS  ON    SURFACE 35 

HYPERBOLOIDS  OF  ONE  NAPPE,  ETC.    assume  elements  through 

GIVEN   POINTS    37 

HYPERBOLOIDS  OF  REVOLUTION  OF  ONE  NAPPE,  assume  Ele- 
ments; SECOND  generations;  construction  of  meridian  curves;  as- 
sume POINTS  ON  SURFACE 37 

HELICOIDS.     OBLIQUE  AND  right;  ASSUME  ELEMENTS;  INTERSECTIONS  WITH 

H  OR  V;  ASSUME   POINTS  ON   SURFACE 38 

APPLICATIONS     39 

PLANES  TANGENT  TO  SURFACES. 


THROUGH  POINTS  ON  THE  SURFACE;  THROUGH   POINTS  OFF  THE  SURFACE;  PAR- 
ALLEL TO  GIVEN  lines;  through  given  lines. 

1.  RIGHT  CYLINDERS  41 

2.  OBLIQUE  CYLINDERS   41 

3.  RIGHT  CONES  43 

4.  OBLIQUE  CONES 43 

5.  HELICAL  CONVOLUTES  43 

6.  HYPERBOLIC  PARABOLOIDS   43 

7.  HYPERBOLOIDS  OF  ONE  NAPPE 43 

8.  HELICOIDS    44 

9.  SPHERES    44 

10.  ELLIPSOIDS    44 

11.  HYPERBOLOIDS  AND  PARABOLOIDS  OF  REVOLUTION  ...  45 
13.     TORI    : 45 

TO  FIND  POINTS  WHERE  LINES  PIERCE  SURFACES 

RIGHT   PRISMS   AND    PYRAMIDS    46 

OBLIQUE  PRISMS  AND  PYRAMIDS 46 

RIGHT  CYLINDERS  AND  CONES 46 

OBLIQUE   CYLINDERS   AND   CONES    46 

WARPED  SURFACES 47 

DOUBLE  CURVED  SURFACES    47 


Viii  CONTENTS 

INTERSECTIONS  AND  DEVELOPMENTS 

I.  INTERSECTIONS  OE  SURFACES  AND  PLANES,  TRUE  SIZES  OF 

INTERSECTION  CURVES,  DEVELOPMENTS,  TANGENT  LINES 
TO  INTERSECTION  CURVES. 

1.  PRISMS  CUT  BY  PI.ANES  AND  DEVFXOPMEINT 48 

2.  PYRAMIDS   CUT  I'Y   PLANES  AND  DICVEILOPMENT 48 

3.  RIGHT  CYLINDERS   CUT  BY   PLANES,  TRUE   SIZE  OF   CURVE,   AND  DEVELOP- 

MENT      49 

4.  OBLIQUE  CYLINDERS  CUT  BY  RIGHT  SECTION   PLANES  AND  DEVELOPMENT  50 

5.  OBLIQUE  CYLINDERS  CUT  BY  OBLIQUE  PLANES  AND  DEVELOPMENT 51 

6.  RIGHT   CONES   CUT   BY  PLANES   AND  DEVELOPMENT 52 

7.  OBLIQUE  CONES  CUT  BY  SPHERES  AND  DEVELOPMENT 53 

8.  OBLIQUE  CONES  CUT  PLANES  AND  DEVELOPMENT 54 

9.  SPHERES   CUT   BY    PLANES 54 

10.  ELLIPSOIDS  CUT  BY  PLANES,  TRUE  SIZE  OF  CURVE  OF  INTERSECTION  ....    55 

11.  PARABOLOIDS  AND  HYPERBOLOIDS  OF  REVOLUTION   CUT  BY   PLANES,   TRUE 

SIZE  OF  INTERSECTION   CURVES,  LINES  TANGENT  TO  CURVES 55 

12.  TORI  CUT  BY  PLANES,  LINES  TANGENT  TO  INTERSECTION  CURVES 56 

13.  SQUARE  RINGS  CUT  BY  PLANES;  TRUE  SIZE  OF  CURVES  OF  INTERSECTION  56 

14.  HYPERBOLIC    PARABOLOIDS    CUT    BY    PLANES 56 

15.  HYPERBOLOIDS  OF  REVOLUTION  OF  ONE  NAPPE,  CUT  BY  PLANES 58 

16.  HELICOIDS  CUT  BY  PLANES,  TRUE  SIZE  OF  CURVES  OF  INTERSECTION  ....  58 

SHORTEST  DISTANCES  BETWEEN  POINTS  ON  SURFACES. 

PRISMS  AND  PYRAMIDS 60 

CYLINDERS  AND  CONES    60 

SPHERES     61 

II.  INTERSECTIONS  OF  TWO  SURFACES. 

1.  TWO    CONES.      INTERSECTION    CURVES,    LINE    TANGENT    TO    SAID    CURVES, 

OR  DEVELOPMENT  OF  ONE  CONE 62 

2.  TWO  CYLINDERS.     INTERSECTION  CURVES,  LINE  TANGENT  TO  SAID  CURVES, 

OR  DEVELOPMENT  OF  ONE  CYLINDER 63 

3.  CONE   AND    CYLINDER.       INTERSECTION    CURVES,    LINE    TANGENT   TO    SAID 

CURVES,  OR  DEVELOPMENT  OF   ONE   SURFACE 64 

4.  GENERAL  INTERSECTIONS  OF  TWO  SURFACES.     INCLUDING  COMBINATIONS 

OF  SINGLE-CURVED,  WARPED,  AND  DOUBLE-CURVED  SURFACES 67 

III.  GENERAL  INTERSECTIONS. 

INCLUDING  THREE  SURFACES 70 

IV.  GENERAL  APPLICATIONS. 

BASED  ON  INTERSECTIONS  AND  DEVELOPMENTS 71 

BLim  PRINT  FIGURE  INDEX 74 


INTRODUCTORY. 

Working  Space.  Adopting  a  12"  x  18''  sheet  with  ^"  border  as  a  standard, 
problems  have  been  divided  into  -1  classes:  (1)  those  for  whose  solution  a  whole 
sheet  is  necessary,  with  the  ground  line  as  shown  in  Fig.  1;  (2)  those  which 
can  be  arranged  two  on  a  sheet,  with  ground  lines  as  shown  in  Fig.  2;  (3)  those 
which  can  be  arranged  four  on  a  sheet,  with  ground  line  as  shown  in  Fig.  3 ; 
(4)  those  for  whose  solution  -J  of  a  sheet  is  necessary,  with  ground  line  as  shown 
in  Fig.  3.  After  the  problem  number,  a  number  is  given  in  brackets,  thus  [1], 
[2]  J  [4],  [8],  to  indicate  whether  1,  2,  4  or  8  such  problems  can  be  worked  upon 
a  standard  sheet.  Where  such  a  number  in  brackets  is  given  in  a  group  heading, 
any  problem  in  this  group  can  be  worked  in  the  space  indicated.  Thus  in  the 
solution  of  problems  in  the  drawing  room  or  at  home  a  standard  sheet  may  be 
used  and  a  number  of  problems  arranged  thereon  in  accordance  with  the  space 
required,  or  each  problem  may  be  solved  upon  a  separate  sheet,  using  standard 
sheet,  half  standard,  quarter  standard,  etc.,  as  the  particular  problem  may  require. 

The  Profile  Plane  and  the  Origin.  Unless  otherwise  stated  the  reference 
profile  plane  is  assumed  to  be  perpendicular  to  the  ground  line  XX  at  the  point 
where  the  latter  cuts  the  right  hand  boundary  line  of  the  working  space.  Thus, 
in  Fig.  4,  the  profile  planes  would  be  assumed  to  be  perpendicular  to  the  ground 
lines  XX  at  the  points  marked  O.  The  right  hand  boundary  line  of  each  working 
space  would  then  represent  the  profile  ground  line.  The  point  O  is  the  origin 
of  co-ordinates  and  from  it  will  be  located  all  points,  as  per  the  paragraph  below. 
Where  it  is  desirable  to  have  the  profile  plane  located  otherwise  for  any  particu- 
lar problem,  or  group  of  problems,  the  distance  of  the  point  O  from  the  right 
hand  boundary  line  of  the  working  space  will  be  given.  Thus,  a  statement, 
P  at  -  5",  would  locate  the  profile  plane  and  origin  O  as  shown  in  Fig.  5. 

Location  of  points  and  right  lines  by  co-ordinates.  Adopting  the  usual 
notation  of  Analytic  Geometry,  a  point  is  located  in  space  by  its  X,  Y  and  Z 
co-ordinates,  these  co-ordinates  respectively  giving  its  distance  from  the  profile, 
the  vertical  and  the  horizontal  planes.  A  right  line,  likewise,  is  located  by  giving 
the  co-ordinates  of  two  of  its  points.  X  distances  are  considered  +  when  to  the 
right  of  the  profile  plane,  and  give  in  a  particular  case  the  distance  from  the  pro- 
file ground  line,  of  the  line  joining  the  H  and  V  projections  of  a  point.  Y  dis- 
tances are  considered  +  when  in  front  of  the  vertical  plane,  and  give  the  distance 
of  the  H  projection  of  a  point  from  the  ground  line.  Z  distances  are  considered 
4-  when  above  the  horizontal  plane,  and  give  the  distance  of  the  V  projection  from 
the  ground  line. 

In  short,  distances  are  positive  zuhen  to  the  right  of  P,  the  front  of  V,  and 
above  H.  Thus  a  point  M(- 4"*,  +  2",  -  3")  is  a  point  4  inches  to  the  left  of  P, 
2"  in  front  of  V,  and  3"  below  H,  therefore  in  the  4th  quadrant.  To  illustrate 
the  complete  problem  notation,  suppose  a  problem  reads : — 

173.  [2]  P  at  -  3-J-".  Find  the  three  projections  of  the  line  joining  the  point 
A(-2",-3",-l")  with  the  point  B(- 4",  +  1",  +  1").  The  lay-out  is  shown 
with  dimensions  in  Fig.  6. 

Location  of  planes.  The  traces  of  a  plane  are  located  by  giving  first  the  dis- 
tance from  O  of  the  point  of  intersection  of  the  traces  with  the  ground  line  XX, 


then  the  angle  which  the  H  trace  makes  with  the  ground  Hne,  then  the  angle 
which  the  V  trace  makes  with  the  ground  line.  Employing  the  usual  trigono- 
metric notation,  angles  measured  counter-clockwise  are  positive,  clockwise  neg- 
ative.   The  plane  T(- 12'',  +  60°,  - -±5°)  is  shown  located  in  Fig.  7. 

In  case  a  plane  T  is  parallel  to  the  ground  line,  both  traces  will  be  parallel  to 
the  ground  line  and  their  point  of  intersection  with  the  ground  line  is  removed 
to  an  infinite  distance  from  O.  The  notation  T(  co,  +  4",  -  3'')  here  indicates 
first  that  the  traces  are  both  parallel  to  the  ground  line,  second  that  the  H  trace 
is  4"  in  front  of  the  vertical  plane,  third  that  the  V  trace  is  3"  below  H.  The 
drawing  board  representation  is  given  in  Fig.  8.  The  notation  S(oo,-3",  oo ) 
indicates  a  plane  S,  parallel  to  and  3''  behind  V. 

Planes  may  also  be  determined  by  giving  the  co-ordinates  of  three  points, 
or  of  two  intersecting  or  parallel  lines  therein.  In  special  cases,  when  the  above 
notation  for  traces  is  not  convenient,  the  traces  may  be  located  by  co-ordinates, 
considering  them  merely  as  right  lines  in  H,  V  and  P  respectively. 

Abbreviations  which  may  be  used. 

alt.  =  altitude.  pt.  =  point, 

const.  =  construction,  quad.  =  quadrant, 

cyl.  =  cylinder.  rt.  =  right, 

dia.  =  diameter,  tang.  =  tangent, 

obi.  =  oblique.  G.  L.  =  ground  line, 

par.  =  parallel.  H  =  horizontal  plane, 

perp.  =  perpendicular.  V  =  vertical  plane, 

proj.  =  projection.  P  =  profile  plane. 
Fig,  =  figure. 


PRELIMINARY   PROBLEMS. 


1.— TO  FIND  THE  H  AND  V  PROJECTIONS  OF  POINTS. 

1.  Point  A  in  1st  quad.,  2"  from  V,  1"  from  H. 

2.  Point  B  in  2nd  quad.,  f"  from  V,  2"  from  H. 

3.  Point  C  in  3rd  quad.,  V  from  V,  1^  from  H. 

4.  Point  D  in  4th  quad.,  |"  from  V,  1^'  from  H. 


5.  Point  E,  in  4th  quad.,  twice  as  far  from  H  as  from  Y. 

6.  Point  F,  in  3rd  quad.,  one-third  as  far  from  H  as  from  V. 

7.  Point  G,  in  2nd  quad.,  equidistant  from  H  and  V. 

8.  Point  H,  in  1st  quad.,  equidistant  from  H  and  V. 


9.     Point  J,  2"  in  front  of  V,  f ''  below  H. 

10.  Point  K,  V  behind  V,  1''  below  H. 

11.  Point  L,  ly  in  front  of  V,  2''  above  H. 


12.  Point  M  in  V,  1^^'  above  H. 

13.  Point  N  in  V,  1"  below  H. 

14.  Point  O  in  H,  V  in  front  of  V. 


15. 
16. 
17. 

18. 


Point  Q,  in  the  ground  line. 

Points  R  and  vS,  both  in  the  ground  line  and  f "  apart. 
Describe  fully  the  location  of  the  pts.  A,  B,  E,  H,  M,  in  Fig.  9. 
Describe  fully  the  location  of  the  pts.  C,  D,  H,  K,  N,  in  Fig.  9. 


19. 

L 

20. 

L 

21. 

L 

22. 

h 

23. 

L 

24. 

L 

25. 

L 

26. 

L 

27. 

L 

28. 

Li 

29. 

Li 

30. 

Li 

31. 

U 

32. 

U 

33. 

u 

34. 

u 

2.— TO  FIND  THE  H  AND  V  PROJECTIONS  OF  LINES. 

ne  joining  pt.  A  in  1st  quad,  with  B  in  2nd  quad, 
ne  joining  pt.  C  in  2nd  quad,  with  D  in  3rd  quad, 
ne  joining  pt.  E  in  3rd  quad,  with  F  in  4th  quad, 
ne  joining  pt.  G  in  4th  quad,  with  H  in  1st  quad. 
ne  joining  pt.  J  in  2nd  quad,  with  K  in  4th  quad, 
ne  joining  pt.  L  in  1st  quad,  with  M  in  3rd  quad. 


ne  joining  pt.  N  in  H  with  O  in  1st  quad, 
ne  joining  pt.  P  in  H  with  Q  in  2nd  quad, 
ne  joining  pt.  R  in  PI  with  S  in  3rd  quad, 
ne  joining  pt.  A  in  H  with  B  in  V. 
ne  joining  pt.  C  in  V  with  D  in  1st  quad, 
ne  joining  pt.  D  in  V  with  E  in  3rd  quad. 


ne  AB,  par.  to  H,  obi:  to  V,  in  1st  quad, 
ne  CD,  par.  to  H,  obi.  to  V,  in  2nd  quad, 
ne  EF,  par.  to  V,  obi.  to  H,  in  3rd  quad, 
ne  GH,  par.  to  V,  obi.  to  H,  in  4th  quad. 


35.  Line  KL  par.  to  ground  line  in  1st  quad. 

36.  Line  MN  par.  to  ground  line  in  8nd  quad. 

37.  Line  OP  par.  to  H  and  V  in  3rd  quad. 

38.  Line  RS  par.  to  H  and  V  in  4th  quad. 


39.  Line  AB  perp.  to  H  in  1st  quad. 

40.  Line  CD  perp.  to  H  in  2nd  quad. 

41.  Line  EF  perp.  to  V  in  3rd  quad. 

42.  Line  GH  perp.  to  V  in  4th  quad. 

43.  Line  KL  lying  in  H  in  front  of  V. 

44.  Line  MN  lying  in  V  below  H. 

45.  Line  OP  in  the  ground  line. 


46.  Line  RS  in  1st  quad,  in  a  plane  perp.  to  the  G.L. 

47.  Line  in  a  plane  perp.  to  ground  line,  joining  pt.  A  in  1st  quad,  to  pt.  B  in 
3rd  quad. 


48.  Describe  fully  the  positions  of  the  lines  AB,  CD,  EF,  KL,  MN,  shown  in 
Fig.  10. 

49.  Describe  fully  the  positions  of  the  lines  GH,  CD,  OP,  KL,  MN,  shown  in 
Fig.  10. 

Intersecting  lines. 

50.  Lines  AB  and  BC  intersecting  at  a  pt.  B  in  the  1st  quad. 

51.  Lines  CD  and  DE  intersecting  at  a  pt.  D  in  the  3rd  quad. 

52.  Line  FG,  par.  to  H  intersecting  line  GH  at  a  pt.  G  in  2nd  quad. 

53.  Line  KL,  par.  to  V  intersecting  line  LM  at  a  pt.  L  in  3rd  quad. 

54.  Line  NO,  par.  to  the  G.L.,  intersecting  line  OP  at  pt.  O  in  4th  quad. 

55.  Line  PQ,  par.  to  the  G.L.,  intersecting  line  QN  at  pt.  Q  in  3rd  quad. 

56.  Line  QR  perp.  to  H,  intersecting  RS,  par.  to  the  G.L. 

57.  Line  TU  perp.  to  H,  intersecting  UV,  par.  to  H. 

58.  Line  YZ,  perp.  to  V,  intersecting  ZM,  par.  to  the  G.L. 

Parallel  lines. 

59.  Parallel  lines,  obi.  to  H  and  V,  through  pts.  A  and  C  in  1st  and  3rd  quads. 
respectively. 

60.  Parallel  lines,  obi.  to  H  and  V,  through  pts.  B  and  D  in  2nd  and  3rd  quads, 
respectively. 

6L     Parallel  lines,  obi.  to  H  and  V,  through  pts.  E  and  G  in  3rd  quad,  and  in  H 
respectively. 

62.  Parallel  lines,  obi.  to  H  and  V,  through  pts.  F  and  K,  in  1st  quad,  and  in  V 
respectively. 

63.  Parallel  lines,  in  3rd  quad.,  par.  to  H  and  obi.  to  V. 

64.  Parallel  lines,  both  par.  to  the  G.L.,  one  in  3rd  quad.,  one  in  H. 

65.  Parallel  lines,  both  perp.  to  H,  both  in  4th  quad. 


3.— TO  DETERMINE  THE  TRACES  OF  PLANES. 

66.  Plane  R,  obi.  to  both  H  and  V. 

67.  Plane  S,  par.  to  H,  and  above  H. 

68.  Plane  T,  par.  to  V,  and  behind  V. 

69.  Plane  U,  par.  to  ground  line,  cutting  across  the  1st  quad.  ' 

70.  Plane  R,  perp.  to  ground  line. 

71.  Plane  S,  perp.  to  H,  obi.  to  V. 
73.  Plane  T,  perp.  to  V,  obi.  to  H. 

73.  Plane  U,  passing  through  the  ground  line. 

74.  Describe  fully  the  planes  R,  S,  U,  W,  Z  shown  in  Fig.  11. 

75.  Describe  fully  the  planes  R,  P,  Q,  T,  Z  shown  in  Fig.  11. 

Parallel  planes. 

76.  Par.  planes,  R  and  S,  obi.  to  both  H  and  V. 

77.  Par.  planes  T  and  U,  perp.  to  H,  obi.  to  V. 

78.  Par.  planes  V  and  W,  perp.  to  V,  obi.  to  H. 

79.  Par.  planes  R  and  S,  perp.  to  ground  line. 

80.  Par.  planes  T  and  U,  par.  to  V. 

81.  Par.  planes  V  and  W,  par.  to  H. 

4.— TO  ASSUME  RIGHT  LINES  IN  GIVEN  PLANES. 

In  plane  S,  Fig.  11  ;  (b)  in  plane  U,  Fig.  11. 
In  plane  S,  Fig.  11  ;  (b)  in  plane  U,  Fig.  11. 
In  plane  W,  Fig.  11 ;  (b)   in  plane  Z,  Fig.  11. 
In  plane  W,  Fig.  11 ;  (b)   in  plane  Q,  Fig.  11. 

5.— REVOLUTION  OF  POINTS  ABOUT  GIVEN  AXES. 

86.  The  pt.  A,  Fig.  12,  is  to  be  revolved  about  the  axis  MX  in  H.  Find  the  2 
projections  of  the  pt.  after  revolution  into  H. 

87.  The  pt.  B,  Fig.  13.  is  to  be  revolved  about  the  axis  RS  in  H.  Find  the  2 
projections  of  the  pt.  after  revolution  into  H. 

88.  The  pt.  C,  Fig.  14,  is  to  be  revolved  about  the  axis  PQ  into  V.  Find  the  2 
projections  of  the  pt.  where  it  falls  into  Y. 

89.  The  pt.  D,  Fig.  15,  is  to  be  revolved  about  the  axis  RS  into  V.  Find  the 
2  projections  of  the  pt.  where  it  falls  into  V'. 

Revolution  of  lines  about  given  axes. 

90.  The  line  AB,  Fig.  16,  is  to  be  revolved  about  the  axis  MN  (par.  to  AB  in  H) 
into  H.     Find  its  2  projections  in  revolved  position. 

91.  The  line  CD,  Fig.  17,  is  to  be  revolved  about  the  axis  PC  (par.  to  CD  in  V) 
into  y.    Find  its  2  projections  in  revolved  position. 

92.  The  line  AB,  Fig.  10,  is  to  be  revolved  into  H  about  its  H  projection  as  an 
axis.     Find  its  2  projections  in  revolved  position. 

93.  The  line  EF.  Fig.  10,  is  to  be  revolved  into  V  about  its  V  projection  as  an 
axis.     Find  its  2  projections  in  revolved  position. 

94.  The  line  EF,  Fig.  10,  is  to  be  revolved  about  the  H  projecting  perpendicular 
of  the  pt.  E  until  it  is  par.  to  V.    Find  its  2  projections  in  revolved  position. 

95.  The  line  OP,  Fig.  10,  is  to  be  revolved  about  the  V  projecting  perpendicular 
of  the  pt.  O  until  it  becomes  par.  to  H.  Find  its  2  projections  in  revolved 
position. 


82. 

(a) 

82. 

(a) 

84. 

(a) 

85. 

(a) 

6.— PROFILE  PROJECTIONS  OF  POINTS. 

The  student  is  referred  to  the  problems  in  Set  I.  After  assuming  a  profile  plane, 
in  each  case,  find  the  profile  projections  of  the  points  given 

96.  In  problems  1,  3,  7,  12. 

7.— PROFILE  PROJECTIONS  OF  LINES. 

The  student  is  referred  to  the  problems  in  Set  2.     Having  assumed  a  profile 
plane,  in  each  case,  find  the  profile  projections  of  the  lines 

97.  In  problems  25,  31,  39. 

98.  In  problems  27,  33,  41. 

8.— PROFILE  TRACES  OF  PLANES. 

The  student  is  referred  to  the  problems  in  Set  3.     Having  assumed  a  profile 
plane  in  each  case,  find  the  profile  traces  of  the  planes 

99.  In  problems  6Cy,  67,  69. 

100.  In  problems  66,  68,  73. 


PROBLEMS  ON  THE  POINT,  LINE  AND  PLANE. 


1.— TO    FIND    THE    POINTS    IN    WHICH    A    GIVEN    RIGHT    LINE 
PIERCES  THE  PLANES  OF  PROJECTION. 

Find  the  pts.  M  and  N  in  which  the  following  lines  pierce  H  and  V 

respectively. 
General.   [8] 

101.  Line  joining  pt.  A(-  2^',  -  1",  -  D  with  B(-  1^",  -  2",  -  1"). 

103.  Line  joining  pt.  C(-3",  +  lY',  +  V')  with  D(- U'', +  i",  +  2"). 

103.  Line  joining  pt.  EC-  3",  -  2'',  +  J")  with  F(-  1^",  -  ^'',  +  ID- 

104.  Line  joining  pt.  G(- 34",  -  l^",  -  1")  with  H(- 1^", +  |",  +  2"). 

105.  Line  joining  pt.  K(- 2f",  +  U",  +  1")  with  L(- U".  +  i".  -  !")• 

Lines  par.  to  H  or  V.    [8] 

lOG.     Line  joining  pt.  A(-  3",  -  f ',  -  1")  with  B(-  U/',  -  1^',  -  1"). 

107.  Line  joining  pt.  C(- 3'',  +  1^',  + f")  with  D(- iV',  +  i",  +  f")- 

108.  Line  joining  pt.  E(-  3",  -  V' ,  -  UJ')  with  F(-  U'^-V,  +  H")- 

109.  Line  joining  pt.  G(- 2f' ,  ^  1",  +  1")  with  H(- 1^",  +  1",  -  2'')- 

110.  Line  joining  pt.  K(-  3",  -  Y',  +  f")  with  L(-  1|'',  -  2",  +  f")- 

Lines  in  plane  par.  to  P.  [8]    P  at  -  2". 

111.  Line  A(-l^-ir,-U")   B(- 1",  -  T,  -  D- 

112.  Line  0(0'',- U",  +  i")  D(0^-i",-l|'O. 

113.  Line  ECO'', -4", -ID  F(0",  +  H",  +  If')- 

Find  the  H,  V  and  P  piercing  points  (M,  N,  O)  of  lines.    [8]    P  at  -  2". 

114.  Line  A(-H",-D-2")   B(- r.  -  D -i")- 

115.  Line  C(-ir,-r',  +  li'0  D(-y',  +  r',  +  |'0- 

116.  Line  E(-  D  -h  1",  +  f)  F(+f",  -  f',  -  Y')- 

Find  the  3  projs.  of  the  P  piercing  point  (O)  of  given  lines.    [8]    P  at  -  1|-". 

117.  Line  A(-ir;-l", -D   B(- D  +  1",  -  i"). 

118.  LineC(-2",-^",-li'0  DC-^-D-lD- 

119.  Line  E(-2'',  0'^ -4")  F(-D0'',-li'O. 

Find  the  points  where  the  foei.owinc  lines  pierce  a  plane  parall5;l  to  h 

and  below  h.    [8] 

120.  Line  M(-2f' ,  +  r',  +  rO  N(- 1'',  +  i'',  -  2''). 

121.  Line  O(-3",-D-2'0  P(- 1^', -^^  +  H'')- 

122.  Line  R(-3^-D-l|'')  S(- D  -  1",  -  i'O- 

find  the  projections  of  the   lines   which   pierce  the  PLANES  OF  PROJECTION 

RESPECTIVELY,  AS  FOLLOWS.     [8] 

123.  V  at  A(-  V',  0",  -  I'O,  H  at  B(-  1^' ,  -  2",  0"). 

124.  V  at  C(-  3'',  (r,  -  H'O,-  H  at  D(-  ij",  +  2",  O'')- 

135.     Pat -2".    H  at  E(-1D-D0"),  Pat  F(0^-li^-rO. 


APPLICATIONS. 

126.  [4]  On  the  plans  for  the  mill  building'  shown  in  Fig.  18,  the  line 
A(-3^--f' ,-2r)  B(-3'',--r,-J'0  gives  the  direction  of  the  center  line  of 
a  vertical  pipe.  Find  the  right  end  view  (P  projection)  of  the  building  and  the 
3  projections  of  the  points  M  and  N  where  this  line  pierces  the  floor  planes  F 
and  S. 

137.  [4]  On  the  plans  for  the  mill  building  shown  in  Fig.  18,  the  line 
C(-2'',-i'',-li'0  D(-2",-2",-l-l")  gives  the  direction  of  the  center  line 
of  a  telephone  wire.  Find  the  right  end  view  (P  projection)  of  the  building  and 
the  3  projections  of  the  points  K  and  L  where  this  telephone  wire  pierces  the 
walls  parallel  to  V. 

128.  [4]  On  the  plans  for  the  mill  building  shown  in  Fig.  18,  the  line 
E(-r',-f' ,-U'')  F(-Y\-^'',~iy)  gives  the  direction  of  the  center  line 
of  a  shaft  for  an  outside  rope  drive.  Find  the  right  end  view  (P  projection)  of 
the  building  and  the  3  projections  of  the  point  R  where  this  line  pierces  the  right 
hand  end  of  the  building. 

129.  [8]  In  Fig.  32,  find  the  point  X  where  the  tow-rope  MB  pierces  the  hori- 
zontal plane  (extended)  of  the  tow-path. 

130.  [8]  In  Fig.  32,  find  the  point  where  the  tow-rope  MB,  if  extended,  would 
pierce  a  vertical  plane  through  the  rear  end  of  the  canal  boat. 

131.  [2]  G.  L.  par.  to  long  side  of  space.  P  at  -  4".  Suppose  that  the  telegraph 
wire  in  Fig.  30  were  to  be  extended  in  the  line  PW.  Find  (1)  the  profile  projec- 
tion (right  end  view)  of  the  line  and  factory  building,  (2)  the  3  projections  of 
the  point  X  where  the  line  PW  would  pierce  the  horizontal  plane  of  the  floor 
ABC,  of  the  point  Y  where  it  would  pierce  the  plane  of  the  rear  wall  parallel 
to  ABFE,  and  of  the  point  Z  where  the  line  PW  would  pierce  the  plane  of  the  left 
vertical  end  wall  of  the  building. 

132.  [2]  G.  L.  par.  to  long  side  of  space.  P  at  -4".  Suppose  that  the  tele- 
graph wire  in  Fig.  30  were  to  be  extended  in  the  line  PW.  Find  (1)  the  profile 
projection  (right  end  view)  of  factory  building  and  line  PW,  (2)  the  3  pro- 
jections of  the  point  X  where  the  line  PW  would  pierce  the  horizontal  plane  of 
the  roof  of  the  building,  of  the  point  Y  where  the  line  PW  would  pierce  a  verti- 
cal partition  plane  half  way  between  and  parallel  to  the  front  and  rear  walls  of 
the  main  part  of  the  building,  of  the  point  Z  where  the  line  PW  would  pierce  the 

plane  of  the  left  end  wall  of  the  building. 

» 

2._TO   FIND   THE   TRUE   LENGTH   OF  A   RIGHT   LINE  JOINING 
TWO  GIVEN  POINTS  IN  SPACE. 
General.  [8] 

133.  Line  joining  pt.  A(-  3|'',  -  1^'',  -  H")   with  B(-  If,  -  i",  -  Y'). 

134.  Line  joining  pt.  C(-3'',  -  2",  +  i")  with  D(-1Y',-V'>  +  '^D- 

135.  Line  joining  pt.  E(-  S^',  -  U",  -  1")  with  F(-  1^'',  +  Y',  +  I'O- 

136.  Line  joining  pt.  G(-  3'',  -  Y',  !-  I'O  with  H(-  1^',  -  1^.  -  I'O- 

137.  Line  joining  pt.  K(- 2f'',  +  U",  f  I'O  with  L(- H",  +  1",  -  I'O- 

Lines  joining  points  in  H,  V  or  P.    [8] 

138.  Line  joining  pt.  A(-  3",  0",  -  I'O  with  B(-  1^',  -  2",  0")- 

139.  Line  joining  pt.  C(-  3'',  0'',  -  1^")  with  D(-  1",  +  2",  O'O- 

140.  P  at  -  v.    Line  joining  pt.  E(-  If",  0",  -  1")  with  F(0",  +  H'',  -f  i'O- 


Lines  in  plane  perp.  to  G.  L.    [8]    P  at  -  2". 

141.  Line  joining  pt.  M(0",  -  i",  -  IV')  with  N(0",  -  1",  -  i"). 

142.  Line  joining  pt.  0(0",  -  V',  -  I'O  with  P(0'',  +  1|'',  -  li"). 

143.  Line  joining  pt.  R(0",-1|'',  f|")   with  S(0",  -  1'',  -  14"). 

TO  Find  a  point  on  a  given  line;  a  given  distance  erom  one  End.    [8] 

144.  Find  pt.   in   Hne   P(- 3^",  -  H",  -  1^")    R(- l^"^  _  i",  _  i/')    which   is   f" 
from  R. 

145.  Find  pt.  in  Hne  M(-  3",  0",  -  1")  N(-  H",  -  3",  0")  which  is  H"  from  M. 

146.  Find  pt.   in   Hne   K(- 3|",  -  U",  -  1")    L(- li",  +  T- +  I'O    which   is   1" 
from  K. 

TO  find  the  angle  a  given  right  line  makes  with  h  and  v.    [8] 

147.  Line  M(-  3|",  -^",  0")  N(-  1^",  -  2",  -  1"). 

148.  Line  0(-2f",  +  l|",  +  l")   P(- 1^',  +  |",  -  1"). 

149.  Line  D(-  3",  +  i",  -  l-J")  E(-  1",  H  2",  -  i"). 

TO  FIND  THE  PROJECTIONS  OF  A  RIGHT  LINE  MAKING  GIVEN  ANGLES 
WITH   H  AND  V.     [8] 

150.  Find  the  projections  of  a  Hne  through  the  pt.  0(- 2|",  +  1^",  +  1")  mak- 
ing 30°  with  H  and  45°  with  V. 

151.  Find  the  projections  of  a  Hne  through  the  pt.  M(- 2",  -  f",  -  f")  making 
45°  with  H  and  22^°  with  V. 

152.  Find  the  projections  of  a  Hne  through  the  pt.  P(- 2^",  -  1^",  -  |")   mak- 
ing 22i°  with  H  and  30°  with  V. 

TO  FIND  THE  3   PROJECTIONS  OF  A  POINT   M  ON   A  GIVEN   LINE,  EQUIDISTANT  FROM 

BOTH   H  AND  V.     [8]     P  AT  -  1^". 

153.  Line  joining  A(-  1|",  -  Y',  -  2")  with  B(-  I",  -  1",  -  i"). 

154.  Line  joining  C(- If",  -  1",  -  U")  with  D(- I",  +  2",  +  i"). 

155.  Line  joining  E(- 2",  -  J",  -  i")  with  F(-^",  -  H",  -  i"). 

APPLICATIONS. 

156.  [4]   Find  the  true  length  of  the  hip  rafter  AC  of  the  cottage  in  Fig.  19. 

157.  [4]   Find  the  true  length  of  the  valley  rafter  MN  of  the  cottage  in  Fig.  19. 

158.  [4]   Find  the  distance  from  the  point  B  in  the  ridge,  to  the  point  P,  Fig.  19. 

159.  [4]  F'ind  the  angle  which  the  rafter  BD  makes  with  a  horizontal  plane 
through  CD,  Fig.  19. 

160.  [4]  Find  the  angle  which  the  hip  rafter  AC  makes  with  a  horizontal  plane 
through  CD,  Fig.  19.' 

161.  [2]  In  the  cap  for  a  ventilator  shaft  shown  in  Fig.  20,  find  the  length  of 
the  intersection  between  the  planes  A  and  B ;  also  the  distance  from  the  point  N 
on  the  ridge  of  the  ventilator  cap  to  the  point  O. 

162.  [8]    Find  the  length  of  the  tug-of-war  rope  in  Fig.  31. 

163.  [2]  Find  the  lengths  of  the  guy  ropes  AD,  and  CD  of  the  boom  derrick 
shown  in  Fig.  27  and  the  angle  which  CD  makes  with  a  horizontal  plane  through 
C.  Find  also  the  length  of  the  boom  FE  and  the  true  distance  between  the  points 
A  and  C. 


ro 

164.  [2]  Find  the  lengths  of  the  guy  ropes  BD  and  AD  of  the  boom  derrick 
shown  in  Fig.  27,  and  the  angle  which  BD  makes  with  a  horizontal  plane  through 
B.  Find  also  the  distance  from  the  end  of  the  boom  F  to  the  top  of  the  mast  D, 
and  the  straight  line  distance  from  E  to  B. 

165.  [8]  In  the  lamp  shade  of  Fig.  21,  find  the  length  of  the  piece  E  used  on 
the  edge  between  the  planes  A  and  B  ;  also  the  distance  from  the  point  E  to  the 
point  H  of  the  base  of  the  shade.    Scale,  1"  =  I'-O". 

166.  [4]  In  the  skew  bridge  of  Fig.  28,  find  the  length  of  the  members  EI  and 
f ID  of  the  portal  bracing ;  also  find  the  distance  from  the  point  C  to  the  point  B, 
and  from  the  point  M  to  L. 

167.  [4]  In  the  .skew  bridge  of  Fig.  28,  find  the  length  of  the  members  IF 
and  CG  of  the  portal  bracing ;  also  find  the  distance  from  the  point  A  to  D  and 
from  O  to  K. 

168.  [8]   Find  the  length  of  the  tow-rope  MB  of  Fig.  32. 

169.  [4]  What  is  the  length  of  the  telegraph  wire  PW  of  Fig.  30.  How  much 
farther  is  it  from  P  to  the  point  F  of  the  roof.     Scale,  1''  =  20'. 

170.  [8]  Two  stations  A(- 28',  +  14',  +  9')  and  B(- 10',  +  4',  +  1')  are  to  be 
connected  by  a  telegraph  line.  Find  the  shortest  line  that  could  be  constructed. 
Scale,  i"  =  1'. 

171.  [8]  The  center  line  of  a  straight  tunnel  enters  a  hill  at  a  point 
E(-  50',  -  25',  -  90')  and  comes  out  at  a  point  F(-  170',  -  80',  -  35')  on  the  other 
side.  Find  the  length  of  the  tunnel  and  its  grade,  that  is,  the  angle  it  mak'es 
with  the  horizontal  plane.     Scale,  1"  =  50'. 

172.  [8]  Two  wireless  telegraph  stations  are  located  at  the  points 
W(- 500', +  800', +  400')  and  Wi(- 1800',  +  200',  +  400')  respectively.  How  far 
are  they  apart?    Scale,  1"  =  500'.  • 

3.— TO  PASS  A  PLANE  THROUGH  3  GIVEN  POINTS.    [4] 

173.  Pass  plane  through  points  M(- 4^",  -  y,  -  1")    N(- 4", -^",  -  1^")   and 

p(-3r',-ir,-r). 

174.  Pass   plane   through   points   A(- 6",  -  i",  -  1^")    B(- 4",  -  1",  -  i")    and 
C(-U",  +  l",  +  r). 

175.  Pass  plane  through  points  E(- 6",  -  f,  -  1|")    F(- l^",  +  1^",  -  |")   and 
G(-2",0",-U"). 

TO  PASS  A  PI^\NE  THROUGH   TWO   INTERSECTING  LINES. 

General.    [4] 

176.  Lines  A(- 6",  +  1",  0")    B(- 4",  -  2",  +  1")   and  BC(- 2|",  +  i",  -  H"). 

177.  Lines  D(- of ,  +  |",  -  U")  E(- 3^",  +  If, -i")  and  EF(-4f ,  0",  +  f ). 

178.  Lines  G(- 6",  0",  -  2")   H(- 4",  -  2",  -  |" )   and  HK(- U",  0",  0"). 

One  line  parallel  to  H  or  V  or  G.  L.    [4] 

179.  Lines  A(- 5",  -  1",  -  2")   B(- 3",  -  1",  -  f )  and  BC(- 4",  0",  0"). 

180.  Lines  D(-  6".  -  If,  -  |")  E(-  3",  +  1|",  -  ^")  and  EF(-  2^",  0",  +  2"). 

181.  Lines  G(- 6",  -  f ,  -  If )   H(- 3",  -  1",  -  If )   and  HK(- 4",  -  If ,  0"). 


II 

One  line  in  plane  parallel  to  P.    [4] 

182.  P    at    -3".      Line    A(-1^0",-ir)     B(-l",-l^-r)     and    the    line 

183.  Line  D(- 5-^,+ J-'',  -  l.D    Z(- 3V\  + 1^',- V)    and   the  intersecting  line 

EF(-3r,-ir,+r)- 

184.  Lines  A(- 3", -r  1'',  -  2'')   B(- 3",  +  1'',  -  I")  and  BC(- 5'',  0",  -  1"). 

TO  PASS  A   PLANE  THROUGH   TWO   PARALLEL  LINES.       [4] 

185.  Line  A(-  5'^  -  U",  f  V)   B(-  3",  0",  -  1.^0   and  line  par.  to  AB  through 

c(-2-,  +  ir,-r)- 

186.  Line  D(- 5'',  +  r',  -  4")  E(- 4",  -  r,  -  1")   and  hne  par.  to  DE  through 
F(-4".-H",  0"). 

187.  Lines  par.  to  G.  L.  through  K(- 6", -]-",- 2|")   and  M(- 4'',  +  f",  -  y'). 

TO  PASS  A  PLANE  THROUGH  A  POINT  AND  A  RIGHT  LINE.     [4] 

188.  Right     line     F(- 6'',  -  1",  -  2")      G(-2V',- 1V\  + V)      and      the     point 
H(-4|^  +  ]",  +  ^"). 

189.  Right  line  parallel  to  ground  line  through  D(- 6",  -  ^",  -  l^")   and  point 

E(-l",-l:r,  0"). 

190.  Right    line    A(- 3'',  +  1'',  -  2")     B(- 3'',  +  1",  -  i'')     and    a    given    point 
C(-5",-r,-l"). 

APPLICATIONS. 

191.  [4]  Assuming  a  ground  line  somewhere  between  the  two  views  of  the 
cottage  in  Fig.  19,  find  the  traces  of  the  roof  planes  U,  V  and  W. 

192.  [4]  Assume  a  convenient  ground  hne  in  Fig.  39  and  find  the  traces  of  the 
roof  planes  R,  S,  T  and  X. 

193.  [2]  Using  the  derrick  of  Fig.  27,  find  (1)  the  plane  of  the  2  guy  wires 
DC  and  DB,  (2)  a  plane  passed  through  the  pts.  E,  C  and  D,  (3)  the  plane  of  the 
boom  FE  and  the  point  C. 

194.  [4]  In  the  skew  bridge  of  Fig.  28,  find  (1)  the  traces  of  the  plane  of  the 
portal,  that  is  the  plane  ABDC,  (2)  the  traces  of  a  plane  through  the  points 
C,  D  and  L. 

195.  [4]  In  the  skew  bridge  of  Fig.  28,  find  (1)  the  traces  of  the  plane  of  the 
lines  AC  and  CD,  (2)  the  traces  of  a  plane  through  the  2  parallels  MN  and  DJ. 

196.  [2]  In  the  Gondola  Car  of  Fig.  23,  the  several  plates  are  shown  by  the 
projections  of  their  bounding  edges.  Having  assumed  a  convenient  ground  line, 
find  the  traces  of  the  planes  of  the  plates  A,  C,  D  and  the  door  plate. 

197.  [2]  G.  L.  par.  to  long  side  of  space.  P  at  -4".  In  Fig.  30,  find  the 
three  traces  of  the  front  wall  of  the  wing  of  the  factory  bi\ilding ;  also  the  traces 
of  the  plane  of  the  three  points  P,  W  and  B. 

198.  [4]  Three  vertical  wells  are  driven  at  points  A(- 50',  + 17',  +  53') 
B(- 95', +  35', +  44')  and  C(- 135',  +  15',  +  25')  of  a  hill  side,  striking  water  at 
depths  of  32  ft..  20  ft.  and  22-^-  ft.  respectivelv.  Find  the  traces  of  the  water 
plane.     Scale,  1"  =  20'. 


12  I       . 

4.— TO  FIND  THE  TRUE  SIZE  OF  THE  ANGLE  BETWEEN  TWO 
GIVEN  INTERSECTING  LINES  AND   TO   FIND   THE  PRO- 
JECTIONS OF  ITS  BISECTOR. 
General.    [4] 

199.  Angle   A(- 2-^^  -  1^'', +  *'')    B(- 6'',  -  1",  -  2")    C(-4^^  +  1'', +  i")    be- 
tween lines  AB  and  BC. 

200.  Angle    D(- 3^',  -  H",  +  f'O     E(- 5f',  +  i'',  -  H")     F(- 3|'',  +  1^'', -I'O 
between  lines  DE  and  EF. 

201.  Angle    G(- 4'',  -  2",  +  1'')     H(- 2|",  +  i",  -  1|'')     K(- 6",  +  1'',  0'')     be- 
tween GH  and  HK. 

One  side  parallel  to  H  or  V,  or  G.  L.    [4] 

202.  Angle  A(-7",-r,-l")    B(- 6",  -  r,  -  1")    C(- 4",  -  f, +  |'')    between 
lines  AB  and  BC. 

203.  Angle    D(-3^-r',-D     E(- 5'',  -  1'',  -  2'')     F(-4^0",0'0     between 
lines  DE  and  EF. 

204.  Angle    G(- 6''.  -  If ',  -  |")     H(- 3",  +  If'',  -  f ')     K(- 2^",  0",  +  2")     be- 
tween lines  GH  and  HK. 

One  side  perpendicular  to  H  or  V.    [4] 

205.  Angle   A(- 3'',  ^  H'',  -  2'')    B(- 3",  +  If ',  -  f ')    C(- 5'',  -  f ',  -  If ')    be- 
tween AB  and  BC. 

206.  Angle    D(- 5^, +  f ',  +  1")     E(- 5f ',  +  2",  +  1'')     F(- 4'', -f  f ',  + 2")     be- 
tween DE  and  EF. 

207.  Angle  G(-6^  +  r',  +  fO     H(- 6",  +  1",  +  2")     K(- 4",  -  If,  -  If ')    be- 
tween GH  and  HK. 

APPLICATIONS. 

208.  [4]  In  Fig.  19  find  the  true  size  of  the  angle  between  the  hip  rafters  CA 
and  EA. 

209.  [4]  In  Fig.  19,  find  the  true  size  of  the  angle  between  the  hip  rafter  EA 
and  the  ridge  AB.  Also  find  the  projections  of  a  line  on  the  roof  bisecting  this 
angle. 

210.  [4]  In  Fig.  19,  find  the  true  size  of  the  angle  between  the  two  valley 
rafters  NM  and  MP. 

211.  [2]  In  the  ventilator  cap  of  Fig.  20,  find  the  angle  between  KM  and  MN. 
Also  find  the  projections  of  a  line  in  the  plane  D  which  bisects  this  angle. 

212.  [4]  In  the  roof  of  Fig.  29,  find  the  angle  between  the  hip  rafters  10-14 
and  11-14,  and  the  projections  of  the  line  bisecting  that  angle. 

213.  [4]  In  the  roof  of  Fig.  29,  find  the  angle  between  the  lines  1-7  and  1-5 
and  the  projections  of  the  bisector  of  this  angle. 

214.  [2]  In  Fig.  27,  find  the  angle  between  the  derrick  guy  wires  BD  and  CD. 
If  a  third  guy  wire  be  located  so  as  to  bisect  this  angle  find  its  projection. 

215.  [2]  In  the  derrick  of  Fig.  27,  find  the  angle  between  the  boom  FE  and 
the  mast  DE.  The  b(3om  is  to  be  raised  to  such  a  position  as  to  bisect  the  present 
angle  FED.    Find  its  projections  when  in  such  a  position. 

216.  [4]  In  the  skew  bridge  of  Fig.  28,  find  the  angle  between  the  end  post 
AC  and  the  portal  strut  CD.  In  repairing  the  portal  bracing,  a  brace  was  run 
from  the  point  C  to  the  end  post  BD  so  as  to  bisect  the  angle  ACD.  Find  its 
projections. 

217.  [4]  In  the  skew  bridge  of  Fig.  28,  find  the  angle  between  the  portal  strut 
CD  and  the  diagonal  DL.    Also  find  the  projections  of  the  bisector  of  this  angle. 


13 

218.  [4]  A  canal  boat  B(- 4|'',  +  2^',  + 1'')  is  towed  along  a  canal  bv  two 
mules  M(-3",  +  U",  +  2'0  and  k(- Q¥\  +  V',  +  ¥'),  one  on  each  bank  at  differ- 
ent elevations.  Find  the  angle  between  the  ropes.  Assuming  that  the  mules 
exert  equal  forces,  the  course  of  the  boat  will  be  shown  by  the  bisector  of  the 
angle  between  the  tow  ropes.     Find  this  course. 

5.— TO   ASSUME   CERTAIN   LINES   IN   GIVEN   PLANES. 

Piercing  points  given.    [8] 

219.  In  plane  R(~  1",  +  150°,  -  120°),  find  line  AB  which  pierces  H  1"  behind  V 
and  V  1"  below  H. 

220.  In  plane  S{- 'SV',  +  3()° ,  +  15°),  find  line  CD  which  pierces  H  1"  behind  V 
and  V  y  above  H. 

221.  In  plane  T(- 2^1",  -  90°, -i  15°),  find  line  EF  which  pierces  H  1^"  in  front 
of  V  and  V  1"  above  H. 

Planes  par.  to  ground  line.    [8]    P  at  -  2". 

222.  Plane  R(  x,  -  ^",  -  1'').     Find  the  3  projections  of  the  line  that  pierces 
H  li"  from  P,'and  Y  Y'  from  P. 

223.  Plane   S(  x,  +  .^",  -  2'').     Find  the  3  projections  of  the  line  that  pierces 
H  IV'  from  P  and  V  V'  from  P. 

224.  Plane  T(x,-1''',  x).     Find  the  3  projections  of  the  line  that  pierces  H 
V'  from  P  and  makes  an  angle  of  00°  with  H. 

Lines  to  make  given  angle  with  one  trace.    [8] 

225.  Plane  R(- 3|'',  +  22^,  -  45°),  find  hne  AB  which  pierces  H  1"  behind  V, 
and  makes  an  angle  of  30°  with  the  H  trace. 

226.  Plane  vS(- 3",  -  120°,  -  45°),  find  line  CD  which  pierces  V  1"  below  H, 
and  makes  an  angle  of  G0°  with  the  V  trace. 

227.  Plane  T(- 3",  +  90°,  -  22^ ),  find  Hne  EF  which  pierces  V  f"  below  H, 
and  makes  an  angle  of  45°  with  the  V  trace. 

Lines  parallel  to  one  trace.    [8] 

228.  In  plane  R(- 1'',  +  150°,  -  120°)  find  line  AB  which  is  par.  to  its  H  trace 
and  li"  below  H. 

229.  In  plane  S(-  3|",  -^  30°,  ^  45°)  find  the  line  CD  which  is  par.  to  its  V  trace 
and  1"  behind  V. 

230.  In  plane  T(- 2f",  -  90°,  +  45°)  find  line  EF  which  is  par.  to  its  V  trace, 
and  1"  in  front  of  V. 

Passing  through  G.  L.  at  equal  angles  with  H  and  V  traces.    [8] 

231.  In  plane  R(- 2",  +  90°,  -  90°)  find  line  AB  which  passes  through  the  G.L. 
and  makes  equal  angles  with  its  H  and  V  traces. 

232.  In  plane  S(-  2^",  -  90°,  +  45°)  find  line  CD  which  passes  through  the  G.L. 
and  makes  equal  angles  with  its  H  and  V  traces. 

233.  In  plane  T(-  3",  +  120°",  -  45°)  find  line  EF  which  passes  through  the  G.L. 
and  makes  equal  angles  with  its  H  and  V  traces. 

Lines  parallel  to  and  at  given  distance  from  trace.    [8] 

234.  Plane  T(-  3^',  +  22r,  -  45°),  find  line  par.  to  the  H  trace  and  V'  there- 
from. 

235.  Plane  S(-  2",  +  90°,  +  135°),  find  Hne  par.  to  the  V  trace  and  1"  therefrom 

236.  Plane  R(- 3'',  -  120°,  -  45°),  find  line  par.  to  the  V  trace  and  1^'  there- 
from. 


14 

One  projection  given,  to  find  the  other.    [8] 

237.  The  point  0{-2",-V',z)  is  a  point  in  the  plane  T(- 3^", +  45°,  -  30°). 
Find  its  unknown  projection. 

238.  The  point  M(-2^  +  l",  z)  is  a  point  in  the  plane  R(- 1'',  +  90°,  +  135°). 
Find  its  unknown  projection. 

239.  The  point  0(-2^'',  y,  -  f')  is  a  point  in  the  plane  S(  oo,  -  1",  -  1|").  Find 
its  unknown  projection. 

240.  The  triangle  A(-  2",  -  V',  z)  B(-  U",  -  If',  z)  C(-  1",  -  i",  z)  lies  in  the 
plane  R(- 3f',  +  45°,  -  30°).     Find  its  V  projection. 

241.  The  triangle  M(-2^  +  l",z)  N(-l^  +  li^z)  0(-3^  +  r',z)  lies  in  the 
plane  S(- 1'',  -  90°,  +  150°).    Find  its  V  projection. 

242.  A  parallelogram  in  the  plane  S(  oo,  - 1",  - 1|'')  has  three  vertices 
A(-  If',  y,  -  V')  B(-  3",  y,  -  i")  and  C(-  3f ',  y,  -  1'').  Find  its  projec- 
tions. 

243.  A  parallelogram  in  the  plane  T(- 1",  + 150°,  - 135°)  has  3  vertices 
M(-3^-f',z)  N(-3f",-r',z)  and  0(- 2",  -  |",  z).  Find  its  projec- 
tions. 

APPLICATIONS" 

AND  LOCATION  OF  GIVEN  FIGURES  IN  GIVEN  PLANES. 

244.  [4]  Plane  T(- 6'',-45°,  + 30°).  Find  the  projections  of  a  1"  square, 
lying  in  the  plane  above  H,  two  of  its  sides  par.  to  the  H  trace  and  when  center 
is  at  a  pt.  which  is  ^"  from  the  H  trace  and  2"  from  the  point  of  intersection  of 
the  traces  at  the  ground  line. 

245.  [4]  Plane  S(-  3'',  +  157|°,  -  90°)  is  one  side  of  a  portable  sheet  iron  cot- 
tage. Find  the  projections  of  a  square  window,  whose  center  is  located  IV'  be- 
low H  and  2"  from  the  vertical  trace  of  the  plane.  The  window  is  If  square, 
to  the  scale  of  the  drawing. 

246.  [4]  In  plane  T(- 3'',  +  120°,  -  135°),  find  the  projections  of  a  circle,  of 
IV  diameter,  lying  in  the  plane  below  H  and  behind  V,  whose  center  is  horizon- 
tally projected  at  (-  4^,  -  1'^  0''). 

247.  [4]  The  inclined  plane  S(- 6",  +  90°, -30°)  is  the  top  of  a  sheet  iron 
vat.  A  conveyor  tube  enters  the  vat  perpendicular  to  the  plane  S.  The  section 
of  the  pipe  is  a  circle  of  2"  diameter,  and  the  center  line  of  this  tube  pierces  the 
plane  at  a  pt.  1''  below  H  and  If  from  the  V  trace,  to  the  scale  of  the  drawing. 
Find  the  projections  of  the  hole  to  be  cut  from  the  plane  to  admit  the  conveyor. 

248.  [4]  The  plane  T(- 6", +  45°,  -  30°)  is  the  north  roof  of  an  observation 
tower  on  a  summer  residence.  A  hole  is  cut  in  this  roof  for  a  fire-place  chimney 
which  is  to  be  built  in  the  form  of  a  regular  hexagonal  prism.  The  horizontal 
projection  of  the  hole  is  therefore  a  regular  hexagon,  and  in  the  particular  draw- 
ing, each  side  of  this  hexagon  measures  1",  with  its  center  at  a  point 
0(-3", -]f ,  z)  in  the  roof  plane,  and  with  2  sides  parallel  to  the  ground  line. 
Find  the  V  projection  and  the  actual  size  of  the  hole. 

249.  [4]  A  hole  3  ft.  square  is  to  be  cut  in  the  roof  plane  U  of  Fig.  19  for  a 
vertical  chimney.  The  hole  is  to  have  2  of  its  sides  parallel  to  EF  and  its  center 
located  at  a  point  4  feet  from  EF  and  FB.    Find  the  projections  of  this  hole. 

250.  [2]  A  circular  steam  pipe  of  2''  diameter  passes  through  the  roof 
T(- 3f ,  +  135°,  -  120°)  of  a  factory  extension,  with  its  center  line  horizontal 
and  perpendicular  to  V,  through  a  point  0(- 6",  y",  -  If )  in  the  roof  plane. 
Find  the  2  projections  of  the  hole  cut  in  the  roof. 

251.  [2]  Ornaments  upon  the  glass  sides  A,  B,  etc.  of  the  lamp  shade  in  Fig, 
21,  are  3''  squares  and  radiating  diagonals,  located  symmetrically  as  shown.  Find 
the  projections  of  the  ornaments. 


^5 

6.— TO  FIND  THE  LINE  OF  INTERSECTION,  AB,  OF  TWO  GIVEN 

PLANES. 
General.    [4] 

253.     Planes  T(- 6^, +  45°,  -  30°)  and  U(- 1'',  f  150°,  -  120°). 

253.  Planes  R(-  6'',  +  67^,  ^- 120°)  and  S(-  3",  +  135°,  +  60°). 

254.  Planes  P(- 6^-45°, +  67i°)  and  Q(- 3^  +  60°,  -  45°). 

Including  planes  perpendicular  to  H  or  V.    [4] 

255.  Planes  W(- 5^^-45°, -I  90°)  and  Q(- 2",  +  135°,  +  150°). 

256.  Planes  R(- 6'',  +  90°,  -  30°)  and  S(- 3^",  +  135°,  +  45°). 

257.  Planes  T(- 5", -90°, -30°)  and  V(- 3",  +  135°,  +  90°). 

Including  planes  parallel  to  ground  line.    [4] 

258.  Planes  W(- 6", -45°, +  45°)  and  Q(  oo,  +  2",  +  1")- 

259.  Planes  R(x,  00,  + ly')  and  S(- 6^  +  30°, +  67i°). 

260.  Planes  T(  00,  + ir',-1")  and  U(  oo,  -  IJ'',  -  y'). 

Including  planes  perpendicular  to  ground  line.    [4]    P  at  -  3'\ 

261.  Planes  R(-3'',  +  30°,-22i°)  and  P. 

262.  Planes  S(- 2'',  -  150°, -45°)  and  P. 

263.  Planes  T(- 3'',  +  90°,  -  30°)  and  plane  U  par.  to  P  and  1"  to  left  of  P. 

Traces  not  intersecting  within  limits  of  drawing.    [4] 

264.  Planes  W(-6",  +  67|°,-30°)  and  Q(- 2^  +  112^°,  -  120°). 

265.  Planes  R(- 4'', -90°, +  30°)  and  S(- 5^^-90°, +  67^°). 

266.  Planes  T(  00,-1", +  2'')  and  U(  00, -1^-2''). 

APPLICATIONS. 

267.  [2]  In  Fig.  20,  find  the  traces  of  the  planes  A  and  B  and  then  find  their 
line  of  intersection.     (It  should  check  with  the  line  MO.) 

268.  [2]  In  Fig.  20,  find  the  traces  of  the  planes  A  and  C  and  then  find  their 
line  of  intersection,  XY. 

269.  [4]  In  the  cottage  of  Fig.  19,  find  the  traces  of  the  planes  R  and  V  and 
then  find  their  intersection.     (It  should  check  with  the  line  MO.) 

270.  [2]   In  Fig.  29,  find  the  intersection  of  the  planes  R  and  X. 

271.  [2]  In  Fig.  29,  determine  the  lines  of  intersection  7-8  and  9-8  between  the 
roof  planes  T,  U  and  S. 

272.  [8]  In  Fig.  32,  find  the  projections  of  the  edges  of  the  banks,  that  is,  the 
intersections  of  the  planes  N  and  S  with  the  horizontal  planes  of  the  tow-path 
and  the  north  bank. 

273.  [8]  In  Fig.  31,  find  the  water  lines  on  the  banks,  that  is  the  intersections 
of  the  banks  N  and  S  with  the  plane  of  the  water. 

274.  [4]  In  the  skew  bridge  of  Fig.  28,  find  the  intersection  of  the  plane  of  the 
portal  ABCD  and  the  plane  determined  by  the  diagonal  members  CN  and  CK. 

275.  [2]  Fig.  34  shows  in  outline  two  dormers  on  the  roof  of  a  country  hotel. 
Find  the  valley  lines  of  the  dormers  (that  is,  the  lines  of  intersection  between 
the  dormer  roof  planes  and  the  planes  of  the  roof). 

276.  [2]  A  vertical  partition  wall  is  to  be  built  in  the  top  story  of  the  gymna- 
sium in  Fig.  29,  through  the  point  M  and  parallel  to  the  hip  rafter  11-14.  Find 
the  lines  in  which  the  partition  wall  will  meet  the  roof  planes  T,  X,  U,  S  and  Q. 

277.  [2]  A  concrete  dock  wall  has  cross-section  and  dimensions  as  shown  in 
Fig.  35.  A  retaining  wall  meets  it  at  an  angle,  as  shown.  Find  all  the  intersec- 
tion lines  of  the  planes  of  these  two  walls,  completing  all  views. 


i6 

378.  [4]  The  plane  T(- 3'',  -  22|°,  +  30°)  is  the  top  surface  of  a  vein  of  coal, 
to  which  an  incHned  conveyor  shaft  is  being  run.  The  floor  of  this  shaft  is  de- 
termined by  the  two  Hnes  A(- 4^^  -  If",  -  l^'O  B(- 7'',  -  r, -i'')  and 
C(-  4^'',  -  Y',  -  3i'0  D(-  7'^  +  V,  -  lY').  Find  the  hne  XY  in  which  the  shaft 
plane  will  cut  the  coal  vein. 

279.  [3]  The  five  planes  Q(  oo,  +  ^",  oo  ),  R(- 1",  -  135°,  + 112^°), 
S(- 8", -45°, +  674°),  T(- 3^'', -90°, +  103-1°)  and  U(- 5|'',  -  90°,  +  77^°)  are 
the  faces  of  a  bridge  pier,  such  as  shown  in  Fig.  44,  whose  base  is  in  H  and  whose 
top  is  shown  2V'  above  H.    Find  the  projections  of  the  edges  of  the  pier. 

380.  [2]  The  plane  T(- 6'',  -  60°, +  45°)  is  a  side  plane  of  a  conveyor  hopper. 
A  wooden  shute  feeding  into  this  hopper  has  one  of  its  sides  determined  bv  the 
lines  A(-U'',-1^-1J'')  B(-4^-^^-2i")  and  a  line  through  the  point 
C(-3f", -If', -f'O  parallel  to  AB.  Find  the  line  in  which  the  hopper  plane 
is  cut  by  the  shute  plane. 

381.  [2]  Construct  the  projections  of  a  regular  square  pyramid  (whose  base  is 
2"  square,  altitude  2")  standing  in  the  1st  quad,  on  H.  Assume  a  convenient 
oblique  plane,  and  find  the  intersection  of  the  plane  with  the  pyramid. 


7.— TO   FIND   THE   POINT  P   IN   WHICH   A   GIVEN   RIGHT   LINE 
PIERCES  A  GIVEN  PLANE. 
General.    [4] 
283.     Find  pt.   where  line  A(- 5", +  |",  0")    B(- 3|",  +  2'',  +  1'')    pierces   plane 
T(-2i'',-135°,  +  150°). 

283.  Find  pt.  where  line  C(- 5",  + V\  +  l")  D(- 3", +  ^",  +  1'')  pierces  plane 
S(-5^  +  120°,  +  45°). 

284.  Find  pt.  where  line  E(- 6",  -  1",  -  i")  F(- 4",  -  1",  -  2")  pierces  plane 
U(- 3", +  1121'', -150°). 

Plane  parallel  to  the  ground  line.    [4] 

285.  Find  pt.  where  line  A(- 6",  -  2",  -  1-i")   B(- 3",  -  4",  +  i")  pierces  plane 

T(x,-ir,  ^). 

286.  Find  pt.  where  line  C(- 6",  -  |",  -  2")    D(- 4",  -  i",  -  ^")    pierces  plane 

S(oo,-l",-ir). 

287.  Find    pt.    where    E(- 6",  +  1|",  -  1^")    F(- 5",  +  1",  -  ^")    pierces    plane 

u(<^,-ir,+r). 

Plane  perpendicular  to  H  or  V  or  both.    [4] 

288.  Find  pt.  where  line  A(- 4",  +  |",  0")  B(- 2^",  +  2",  +  IV')  pierces  plane 
T(-6",-30°,-90°). 

289.  Find  pt.  where  line  C  (-  6",  0",  +  1")  D  (-  3",  -  H",  -  1^")  pierces  plane 
S  (-3|",  +  90°,  +  45°). 

290.  P  at  -3".  Find  pt.  where  line  E(- 3",  +  1",  -  ^")  F(- 1",  -  ^",  -  1") 
pierces  P. 

Plane  given  by  2  right  lines  in  it.    [4] 

291.  Find  the  pt.  P  where  the  line  M(- 5^",  +  1^",  0")  N(-4",  0",  +  H") 
pierces  the  plane  determined  by  the  intersecting  lines  A(- 5^",  +  2",  +  1^") 
B(-4i",  +  l^",  f  11")  and  BC(- Sr',  +  |",  +  ^").  Solve  without  finding 
the  traces  of  the  plane. 

292.  Find  the  pt.  P  where  the  line  M(- 5^",  -  H",  -  If")  N(- 3",  0",  -  i") 
pierces  the  plane  determined  by  the  intersecting  lines  A(- 6",  -  1",  +  ^") 
B(-3",-l",-l")  and  BC(- 6",  +  2",  -  1").  Solve  without  finding  the 
traces  of  the  plane. 


17 

293.  Find  the  pt.  P  where  the  line  M(- 6",  -  1'', -|'0  N(- 34",  +  2'', +  i") 
pierces  the  plane  determined  by  line  A(-  6'',  -  I",  -  ^")  B(-  4'',  -  1",  -  V) 
and  a  line  through  C(- -4", +  -i",  -  2")  parallel  to  AB.  Solve  without  find- 
ing the  traces  of  the  plane. 

APPLICATIONS.^ 

294.  [4]  In  Fig.  IS  a  mill  building  is  shown.  It  is  proposed  to  run  a  telephone 
wire  A(-'64',-2',-10')  B(- 28',  -  22',  -  10')  to  an  office  in  the  second  floor. 
Find  the  point  X  where  this  wire  would  go  through  the  roof. 

295.  [4]  In  the  mill  building  of  Fig.  18,  a  brace  is  to  be  run  to  the  roof  from 
a  point  on  a  heavy  beam  in  the  second  floor.  If  the  center  line  of  the  brace  is 
B(-32',-  lG',-20')  R(-10',-24',-2')  find  its  ends  in  the  roof  and  floor. 

296.  [2]  In  Fig.  34,  find  the  valley  lines  of  the  dormers  by  finding  the  points 
where  the  several  hip  lines  pierce  the  roof  planes,  without  finding  the  traces  of 
the  dormer  roof  planes. 

297.  [2]  Find  the  intersections  of  the  faces  of  the  2  concrete  walls  of  Fig.  35, 
by  finding  where  the  edges  of  the  lower  wall  pierce  the  slope  plane  of  the  higher 
wall. 

298.  [2]  In  Fig.  30,  find  where  the  center  line  PW  of  the  telephone  wire,  if 
extended,  would  pierce  horizontal  planes  through  the  bottom  floor,  top  floor  and 
roof  of  the  factory  building,  also  all  vertical  wall  planes  of  the  wing  and  main 
part. 

299.  [4]  A  sloping  floor  in  a  laboratory  is  given  by  its  wall  intersections 
A(-6i",  +  r,  +  r)  B(-3i",  +  f",  +  lf")  and  BC(- 3^,  +  ir>  +  T  )•  A  steam 
pipe  with  center  line  M(-  5^",  -h  If',  +  1'')  N(-  2f",  0",  h-  1")  passes  through  this 
floor  at  a  point  X.    Find  X. 

300.  [4]  The  plane  S(- GV',  +  90°,  -  30°)  is  the  top  surface  of  a  coal  vein, 
to  which  an  inclined  bore-hole  was  driven  in  the  line  A(- 6",  -  2^",  - 1^") 
B(- 3", -^",  -  :^")  from  the  point  A  on  the  ground.  Find  the  point  X  where 
the  hole  strikes  the  vein,  and  the  length  of  the  hole. 

301.  [4]  The  plane  S(- 6J-",  -  30°,  +  30°)  is  the  top  surface  of  a  coal  vein, 
to  which  a  vertical  bore-hole  was  driven,  from  the  point  A(- 4", +  |",  +  2|"). 
Find  the  point  where  the  drill  strikes  the  coal,  and  the  length  of  the  hole,  if  scale 
is  ]"  =  l'-0". 

302.  [2]  Scale,  1"  =  10'.  Vertical  borings  at  three  points  A(-  67',  +  22',  +  23') 
B(-42',  +  6',  +  20')  and  C(- 26',  + 15',  + 12')  of  a  hiflside  show  an  ore  vein  at 
depths  of  20',  5'  and  11'  respectively.  It  is  proposed  to  run  a  shaft  to  this  vein 
with  center  line  M(- 20',  +  30',  +  27')  N(- 45',  +  10',  0').  How  long  wfll  this 
shaft  be? 

303.  [2]  Construct  the  projections  of  a  regular  square  pyramid  (base  2" 
square,  altitude  3")  standing  on  H  in  the  first  quadrant.  Assume  an  oblique  plane 
and  find  the  intersection  of  this  plane  with  the  pyramid.  (Hint:  Join  the  points 
where  the  edges  of  the  pyramid  pierce  the  given  plane.) 

304.  [2]  Construct  the  projections  of  a  3"  cube  in  the  3rd  quadrant  with  its 
upper  base  in  H.  Assume  a  convenient  oblique  plane  and  find  the  intersection  of 
this  plane  with  the  cube.     (See  hint  in  problem  above.) 

305.  [4]  The  plane  M(- 6^',  -  30°, +  45°)  is  a  mirror  from  which  a  ray  of 
light  L(-5^",  +  l",  +  2f")   I(-3",  +  l",  +  i")  is  reflected  at  a  point  R.    Find  R. 

306.  [2]  Two  vertical  poles  are  located  on  a  hillside  S(- 112',  -  30°, +  45°)  in 
such  position  that  their  center  line  if  extended  would  pass  through  the  points 
A(- 76', +  18', +  24')  and  B(- 32',  ^-48',4  28')  respectively.  Find  the  projections 
and  length  of  a  wire  joining  insulators  on  tops  of  these  poles  which  are  20'  above 
the  points  where  the  poles  enter  the  ground.     Scale,  1"  =  16'. 


i8 

8.— THROUGH  A  GIVEN  POINT  TO  DRAW  A  LINE  PERPENDICU- 
LAR TO  A  GIVEN  PLANE  AND  TO  FIND  THE  DISTANCE 
FROM   THE  POINT  TO   THE  PLANE. 
General.   [4] 

307.  Point  A(-3^-f',-rO  and  plane  R(- 6",  +  30°,  -  45°). 

308.  Point  B(-3^     0",  +  i'O.and  plane  S(- 5",  +  120°, +  45°). 

309.  Point  C(-3'',-l'', -2'')  and  plane  T(- 3'',  +  120°, -150°). 

Plane  parallel  to  the  ground  line.   [4]  P  at  -  3''. 

310.  Point  A(-  V,  -  Y',  -  V)  and  plane  R(  x,  -  V,  -  1^"). 

311.  Point  B(-l^-r',-rO  and  plane  S(  00,  -  2'', +  ^'')- 

312.  Point  C(-r%-i",-n")  and  plane  T(oo,-r',  00). 

Planes  perpendicular  to  H  or  V  or  both.     [4] 

313.  Point  A(-5",  +  Y',  +  lV')  and  plane  R(- 6",  +  90°,  +  30°). 

314.  Point  B(-5'',-^",-l'')  and  plane  S(- 3",  + 90°,  -  135°). 

315.  Point  D(-  1",  -V' ,  -  l-^'O  and  profile  plane  P,  at  -  3'^ 

THROUGH    A    GIVEN     POINT    TO    PASS    A    PLANE;     PERPENDICULAR    TO    A    GIVEN 

PLANE.    [4] 

316.  Point  Ai-^'',-lY',-V')  and  plane  R(- 6",  +  30°,  -  30°). 

317.  Point  B(-2'', -!  1'',+  U'0  and  plane  S(- 3",  -  120°,  -  67^°). 

318.  Point  C(-4^+U",  +  i-")  and  plane  T(-6^  +  90°,  -30°). 

THROUGH  A  GIVEN  LINE  IN  A  GIVEN  PLANE  TO  CONSTRUCT  A  PLANE  T  PERPEN- 
DICULAR TO   THE   GIVEN    PLANE.    [4] 

319.  Line  AB  in  plane  S(- 6'',  -  60°,  +  30°),  which  pierces  H  \\"  in  front  of 
V  and  V  l^'  above  H. 

320.  Line  BC  in" plane  R(- 3",  +  135°,  -  150°),  which  pierces  V  %"  below  H  and 
is  parallel  to  the  H  trace  of  plane  R. 

321.  Line  CD  in  plane  U(- 6", +  45°,  -  45°)  which  is  parallel  to  the  V  trace  and 
y  distant  therefrom. 

APPLICATIONS. 

322.  [4]  A  brace  is  to  be  run  from  an  assumed  point  M  in  the  floor  S  of 
Fig.  18,  perpendicular  to  the  roof  R.    Find  its  center  line  projection  and  its  length. 

323.  [4]  In  the  skew  bridge  of  Fig.  28  find  the  distance  from  the  point  N  to 
the  portal  plane,  determined  by  the  points  ABDC. 

324.  [2]  In  the  derrick  of  Fig.  27,  find  the  distance  of  the  end  of  the  boom  F 
from  the  plane  of  the  two  guy  wires  DB  and  DC. 

325.  [2]  In  Fig.  27,  find  the  distance  of  the  top  of  the  derrick  mast  D,  from 
the  plane  of  the  guv  wire  posts  A,  B  and  C. 

326.  [2]  The  'points  A(- 1^,  +  3^', +  r)  B(- 7^',  +  41",  +  ^D  and 
C(- 5V',  + 1'',  +  3f")  are  in  the  roof  of  a  mill  building.  The  force  due 
to  the  wind  is  normal  to  the  roof  plane.  Draw  through  a  point 
M(- 3^",  +  3f", +  44'')  the  projections  of  an  arrow  which  will  show  the 
direction  of  the  wind  pressure.  Make  the  tip  of  the  arrow  just  touch  the 
roof   plane,   and   find   the   arrow   length. 

327.  [4]  The  plane  F(- 3'',  -  135°,  + 135°)  is  the  plane  of  a  fly  wheel,  whose 
shaft  is  perpendicular  to  F.  Assume  the  projections  of  the  center  line  of  the 
shaft  and  find  the  projections  of  a  point  in  the  center  line  which  is  2"  from  the 
wheel  plane  F. 

328.  [4]   The  plane  T(-  7'',  -  22^°,'+  60°)  is  one  plane  of  the  top  of  a  grain  bin. 
A  shute  runs  into  this  bin  perpendicular  to  the  plane  T,  and  has  a  center-line 
length  of  2   feet.     Assume   its  center  line,  and  then  find  both  ends  thereof. 
Scale,  V  =  1'  -  0''. 


19 

339.  [2]  At  three  points  A(- 40',  +  3^',  +  32')  B(- 23',  +  11',  +  15')  and 
C(- 32^', +  24', +  14')  vertical  borings  strike  coal  at  7',  5'  and  12'.  It  is 
proposed  to  run  an  inclined  shaft  perpendicular  to  the  vein  from  a  point 
M(-  25',  +  12',  +  28').    i"  =  1'  -  0".     Find  the  length  of  this  shaft. 

330.  [2]  Borings  in  a  coal  region  show  coal  at  the  points  A(- 25'.  +  1^',  +  3') 
B(-84',  +  9r,  +  8')  and  C(- 14',  +  3',  +  14^').  Assuming  that  the  surface  of 
the  coal  is  a  plane,  find  the  length  of  the  shortest  shaft  that  can  be  driven  from 
the  point  M(-25',  !  9-i',  ^-11')   to  the  vein.     Scale,  ]"  -  I'-O". 

331.  [2]  Vertical  wells  at  points  A(- 124',  +  7',  +  38')  B(- 70',  +  15',  +  82') 
and  C(- 42',  +  47',  4  61')  on  a  mountain  slope  strike  water  at  depths  of  22',  10' 
and  20'  respectively.  Assuming  that  these  3  points  determine  the  plane  of  the 
water,  what  is  the  direction  and  length  of  the  shortest  pipe  that  could  be  driven 
from  the  point  M(-  103',  +  40',  +  64')  to  strike  water.     Scale,  1"  =  20'. 

9.— TO  PROJECT  A  GIVEN  RIGHT  LINE  UPON  A  GIVEN  PLANE. 

General.    [4] 

333.  Line  A(-4r,-ir,-r)  B(- 3",  -  f",  -  §''),  plane  R(- 6",  +  30°,  -  45°). 

334.  Line  B(-5",  +  2",+  l")  C(- 3",  0",  -  i")  and  plane  S(- 5",  +  120°, +  45°). 

335.  Line  C(- 3",  -  1",  -  2")  D(- 5",  -  1",  -  j"),  plane  T(- 3",  +  120°,  -  150°). 

Plane  parallel  to  ground  line.    [4]    P  at  -  3". 

336.  A(-  3",  -  4",  -  2")   B(-  1",  -  ]",  -1")  and  plane  R(  oo,  -  1",  -  1^"). 

337.  Line  BC- 3", +-J",  +  U")   C(- 1",  -  1",  -  1")   and  plane  S(  oo,  -  2", +  1"). 

338.  Line  C(- 3",  -  U",  -  J")    D(- 1",  -  i",  -  U")    and  plane   T(  oo,  -  1",  oo  ). 

Plane  perpendicular  to  H  or  V  or  both.    [4] 

339.  Line  A(- 5",  +  i",  +  li")  B(- 2^", +  2",  +  1"),  plane  R(- 6",  +  90°,  + 30°). 

340.  .Line  B(- 5'',--Y',- 1")  C(- 3",  -  H",  -  1"),  plane  S(- 3",  +  90°,  -  135°). 

341.  Line  C(-  3",  +  i",  +  2")  D(-  1",  -  1",  -  U")  and  profile  plane.     P  at  -  3". 

TO  PROJECT  GIVExNT   FIGURES  UPON   GIVEN    PLANES. 

342.  [4]   Find    the    projection     ABC    of    the    triangle     M(- 6^",  +  2",  +  If) 

N(-5",  +  2",  +  2")   0(-5i",  +  f",  +  l")   upon  plane  R(- 6",  -  45°,  +  45°). 

343.  [4]   Find     the     projection     ABC     of     the     triangle     M(- 5",  - 1|",  - 1") 

N(-  3",-  H",-  1")  0(-  5|",-  yV  U")  upon  plane  S(-  3",+  135°,-  135°). 

APPLICATIONS. 

344.  [4]  Asuming  that  rays  of  incident  light  are  perpendicular  to  the  mirror 
plane  M(- 7J",  -  45°,  J- 30°),  find  the  reflection  of  a  rectangular  frame  ABCD 
of  which  A(-4",-^-f'^  +  l")  B(-4",  + If",  +  1")  and  BC(- 4",  +  If",  0")  are 
2  sides. 

345.  [41  The  plane  B(- 7",  +  90°,  -  30°)  is  the  top  plane  of  a  refuse  bin.  A 
triangular  shute  runs  into  the  bin  perpendicular  to  plane.  If  M(-  3",  -  1^",  -  1^"), 
N(-3",-2", -i")  and  0(- 3",  -  1",  -  J")  are  respectively  points  in  the  3  edges 
of  this  shute,  find  the  hole  to  be  cut  from  the  plane  B. 

346.  [4]  An  engineer's  draughtsman  has  assumed  plane  T(- 7^",  +  30°,  -  90°) 
as  his  vertical  plane  of  projection,  upon  which  he  has  drawn  the  projection  of  a 
triangle  A(- 3f",  -  f",  -  i")  B(- 3",  -  If",  -  2")  C(- 5-1",  -  1",  -  U")-  Find 
the  projections  of  his  projection. 

347.  [4]  The  plane  T(- 2",  +  150°,  -  150°)  is  a  roof  plane  on  a  loading  shed 
for  cars  at  a  coal  mine.  A  flat  belt  conveyor  is  to  pass  through  this  roof  and  per- 
pendicular to  it.     If  A(-5f",-l",-f")   and  B(- 6^",  -  1^",  -  li"-)   are  points 


20 

in  the  two  edges  of  the  belt  respectively,  find  the  width  of  hole  to  be  cut  in  the 
roof  to  just  allow  the  conveyor  belt  to  pass  through.    Let  V  =  1'. 

348.  [8]  An  inclined  mine  shaft  is  to  be  run  perpendicular  to  a  coal  vein 
C('-7V;,  +  90°,-60°).  If  the  points  M(- 3", -i'',  -  2'')  N(- 3'',  -  If',  -  3") 
0(-3'',-|",-^'')  V{-^",-iy\-Y')  are  the  corners  of  the  shaft  entrance, 
find  the  true  shape  and  size  of  the  intersection  of  the  shaft  and  the  vein,  if  1"  rep- 
resents 10', 

10.— THROUGH  A  GIVEN  POINT  TO  PASS  A  PLANE  T,  PERPEN- 
DICULAR TO  A  GIVEN  RIGHT  LINE. 
General.    [4] 

349.  Point  A(-4y',-H'',-l'0.  Hne  M  (- 5|",  -  2",  -  U'')  N(- 4",  -  *",  0"). 

350.  Point  B(-4", -r',-r'),  line  0(- 5",  +  1'',  -  i")   P(- 3",  -  U",  -  H"). 

351.  Point  C(-4",-^",-|")  and  line  R(- 5",  -  1^", +  ^'')  S(- 3",  0",  +  2"). 
Line  parallel  to  H  or  V  or  both.    [4] 

353.     Point  A  (-4^  +  1",  4  1")  and  line  M(- 6",  -  2",  -  f ')  N(-3^0,-f')- 

353.  Point  B(-4",  +  l",+  r')  and  line  0(- 6'',  -  2",  +  ^")   P(- 3",  -  2",  -  I'O. 

354.  Point  C(-  4",  -  1",  -  lY')  and  line  R(-  6",  +  2",  +  1")  S(-  3",  +  2",  +  1"). 

Line  in  plane  perpendicular  to  ground  line.    [4]    P  at  -  3". 

355.  Point  A(-ir',-r', -r).   line   M(- 1",  -  ^",  -  i")    N(- 1",  -  1".  -  1^")- 

356.  Point  B(-3",-H",4l")  and  line  0(0",  -  2'',  -  1")  P(0",  4-^'',  +  U")- 

357.  Point  C(0'',  -  V,  -  U")  and  line  R(0",  -  1^",  -  ^'0  S(0",  -  \\",  -  2''). 

THROUGH  A  GIVKN  POINT,  TO  CONSTRUCT  A  LINE  PERPENDICULAR  TO  A 

GIVEN   LINE.     [4] 

358.  Point  M(-4J",-ir',--l"),  line  A(- 5^",  -  2",  -  If ')    EC- 4",  -  |",  0"). 

359.  Point  N(-4",  +  1" ,  +  1")  and  line  B(-  6",  -  2'',  -  f ')  C(-  3",  0",  -  V). 

360.  Point  M(-  W',  -  V,  -  V'),  line  C(-  4",  -  f ',  -  i")  D(-  4",  -  1",  -  If ')• 

APPLICATIONS. 

361.  [4]  The  line  A(- 3'',  -  f",  -  2f')  B(-7",-2r,-D  is  the  center  line 
of  a  shaft,  upon  which  a  pulley  is  to  be  located  at  a  point  between  A  and  B,  and 
H"  from  A.    Find  the  plane  of  the  pulley. 

"362.  [4]  C(-  6'',  +  If,  +  ]")  D(-  3",  V  If,  +  If)  is  the  center  line  of  a  water 
pipe  which  enters  a  tank  at  the  point  D.  If  the  side  of  the  tank  is  perpendicular 
to  the  water  pipe,  find  its  traces. 

363.  [4]  The  line  M(- Sf,  -  2f ,  -  2f )  N(-5f, -1",-D  is  the  center 
line  of  a  square  conveyor-casing.  Find  the  projections  of  the  casing,  whose  sec- 
tion is  to  the  scale  of  the  drawing,  a  If  square  with  2  sides  parallel  to  the  H 
plane. 

364.  [4]  A  steam  pipe  E(- 3f ,  +  f ,  +  If )  F(- 7",  +  If ,  +  2f )  enters  a 
boiler,  at  the  point  E,  perpendicular  to  its  head  B.    Find  the  plane  of  B. 

365.  [2]  A  square  1"  x  1"  stick  has  the  line  A(-6", -f, -f ) 
B(-3'',  -  4",  -  3f )  as  center  line.  Find  the  projections  of  the  stick,  if  3  of  its 
side  faces  are  perpendicular  to  H. 

366.  [4]  The  plane  R(- 5f ,  -  30°.  +  30°)  is  a  roof  plane  on  a  country  hotel. 
A  tower  is  to  be  constructed  upon  this  roof,  with  one  side  passing  through  the 
line  A(-6^-^2f  ,  +  lf )  B(- 4f , -h  If ,  +  If )  and  perpendicular  to  the  plane 
R.    Find  the  plane  of  this  side. 

367.  [2]  The  plane  of  one  face  of  a  cube  is  determined  by  the  edge 
A(-6^-r',-2f )  B(-3f  ,-3f  ,-f )  and  a  point  C(-4f ,  -  1",  0").  Find 
the  plane  of  the  other  face  of  the  cube  through  the  edge  AB. 


21 

368.  [4]  A  mine  shaft,  driven  in  line  A (-41",  + 1",  + If")  B(- 6",  +  2",  +  |") 
was  found  to  be  perpendicular  to  a  coal  seam  C  at  the  point  B.    Find  the  plane  C. 

369.  [4]  A  telephone  wire  runs  from  a  point  A(- 6",  +  ly,  +  2")  in  a  pole 
through  a  roof  at  a  point  R(- 3J-",  +  1|",  +  f").  If  the  roof  is  perpendicular  to 
the  wire,  find  the  traces  of  its  plane. 

11.— TO  PASS  A  PLANE,  T,  THROUGH  A  GIVEN  POINT  PARALLEL 
TO  2  GIVEN  RIGHT  LINES.    [4] 

370.  Point  M(-4V',-r\-V')   and  lines  A(- 7",  0",  -  1")   B(- 6",  -  1^",  0") 
and  C(-3",-H",  0")  b(- 1",  -  i",  +  1"). 

371.  Point  N(-  4",  -'y\  -  1")  and  lines  D(-  5",  -  f",  -  1")  E(-  4",  -  U",  -  V) 
and  F(-3",  +  l",--l")  G(- 2",  +  U",  +  1"). 

372.  Point  P(-5",  +  l",  +  l"),  lines  G(- 6", -1",  -  IJ")   H(- 5",  -  U",  -  U") 
and  K(-5",4--i",  +  r)  L(- 2", +  i", +  ^"). 

To    PASS   A    PI,ANF;   THROUGH    ONE   LINE   PARALLEL   TO   ANOTHER.      [4] 

373.  Through  line  A(-  5",  0",  +  1")  B(-  4",  -  U/',  0")  par.  to  C(-  3",  -  1|",  0") 
D(-  1",  -V',  +  1"). 

374.  Through   line    C(- 5", -|",  -  1")     D(- 4",  -  1|", -1")     parallel    to    line 
E(-  3",  +  1",  -  i")  F(-  2",  +  ir,  +  1"). 

375.  Through     line     E(- 6",  -  J",  -  U")      F(- 5",  -  U",  -  U")      parallel     to 
G(-  5",  +  i",  +  4")  H (-  2",  + 1",  i  ^") . 

TO  PASS  A  PLANE  THROUGH  A  GIVEN  POINT  PARALLEL  TO  A  GIVEN  PLANE,  AND  FIND 
DISTANCES    BETWEEN    THE    TWO    PLANES.      [4] 

376.  Pass      plane      through      point     A(- 6",  - 1",  -  ^")      parallel      to      plane 
R(-3",  +  112-r,-150°). 

377.  Pass     plane     through     point     B(- 5",  -  ly,  + 1^")      parallel     to     plane 
S(-5i",  +  90°,  +  45°). 

378.  Pass      plane      through      point      C(- 6",  -  i",  -  2")      parallel      to      plane 
T(x,-1",-U"). 

APPLICATIONS. 

379.  [4]  The  Hne  A(- 35',  +  23',  +  5')  B(- 25',  +  10',  +  17')  and  the  line 
BC(- 17',  +  26',  +  17')  are  the  ridge  line  and  valley  line  of  a  cottage  roof.  The 
point  M(-45',  +  20',  +  17')  is  a  point  in  the  ridge  line  of  a  parallel  roof  T  on  an 
adjacent  cottage.    Find  the  trace  of  the  plane  T.     Scale,  1"  =  10'. 

380.  [4]  A  new  building  was  designed  for  a  lithograph  company  with  a  saw- 
toothed  roof,  one  roof  plane  being  R(- 7",  -  90°,  +  30°).  The  points 
M(-4",  +  U",  H-4")  and  N(- 21",  +  l-i",  +  i")  are  points  in  the  ridges  of  two 
other  roof  planes  parallel  to  R.     Find  these  planes. 

381.  [2]  The  line  A(- 80',  -  30',  -  17')  B(- 20',  -  88',  -  82')  is  the  outcrop  of 
a  bed  of  gypsum  on  a  hillside  and  a  vertical  bore-hole  at  point  C(-  55',  -  83',  -  15') 
showed  the  same  bed  at  a  depth  of  20  feet.  At  a  depth  of  60  feet  at  the  point  C 
another  bed  was  struck  which  is  surmised  to  be  parallel  to  the  first.  Find  this 
plane.     Scale,  1"  =  20'. 

382.  [2]  The  hne  A(- 90',  -  42',  -  21')  B(- 54',  -  24',  -  5')  and  the  line 
C(- 90', -24', -12')  D(-54',-6',  +  4')  determine  the  inclined  roof  of  a  tunnel 
into  a  mountain.  The  point  M(- 48',  -  17',  -  10')  is  in  the  floor  of  the  tunnel, 
which  is  parallel  to  the  roof.    Find  the  floor  plane.    Scale,  1"  =  12'. 

383.  [2]  The  line  A(- 7r,  ^- 3^", +  1")  B(- 5^",  +  1",  + 2f")  and  the  line 
BC(-3|:", +  3|", +  ^")  are  two  lines  in  a  gable  roof  plane.  The  point 
M(-3", +  lf", +  1V')  is  a  point  in  another  gable  plane  T  parallel  to  the  first. 
Find  the  traces  of  T. 


22 

12.— TO  FIND  THE  DISTANCE  FROM  A  GIVEN  POINT  TO  A  GIVEN 

RIGHT  LINE. 
General.    [4] 
384.     Point   P(-4r,-U^-l'0,   Hne   A(- 5^'',  -  2'',  -  H'O    B(- 4'',  -  i",  0"). 
38.5.     Point  M(-  4",  -  V,  -  V')  and  line  C(-  5'',  +  1'' -  i'')  D(-  3"  -  1^'-  IV). 

386.  Point  N(-r\-V\-V)   and  line  K(- 5'',- 1^%  +  ^')   F(- 3'',  0",  +  3"). 

When  line  is  parallel  to  H  or  V  or  both.    [4] 

387.  Point  P(-4", +  1",  +  1")  and  line  A(- 6^,-2'',- ^'')  B(- 3",  0", -^'0- 

388.  Point  M(-4",  +  r',  +  -r')  and  line  C(- 6",  -  2",  4-^)  D(- 3'',  -  2",  -  l"). 

389.  Point  N(-  4",  -  1'',  -  iVO,  line  E(-  6",  -  2'',  -^  1'')  F(-  3^",  +  2",  +  I'O- 

Where  line  is  in  plane  perpendicular  to  G.  L.    [4]    P  at  -  3''. 

390.  Point  P(-ir,-l^-i")  andline  A (- IV  r,-D  B(- 1",  -  1",  -  1|")- 

391.  Point  M(-2",-ir,  +  l")  and  line  C(0",-2",-r)  D(0^ +  1",  +  1^"). 

392.  Point  N(0",  -  1",  -  1^')  and  line  E(0",  -  1V',-D  F(0'',  -  1^'',  -  2"). 

TO  FIND  THE  PROJECTIONS  OE  A  LINE  THROUGH  A  GIVEN  POINT  PERPENDICULAR  TO  A 

GIVEN   LINE.     [4] 

393.  Point   P(-4",-l",-r),   ^ne   A(  -  5",  +  1",  -  T)    B(- 3",  -  H".  -  ID- 

394.  Point  M(-4",  +  l",  +  r')  and  line  C(-6",-2",  +  D  D(- 3",  -  2",  -  l'')- 

395.  Point  N(-r/',-U",  +  l"),  line  E(- 4",  -  2",  -  1")  F(- 4",  +  ^",  +  IJ")- 

TO  FIND  THE  DISTANCE  BETWEEN  TWO  PARALLEL  LINES.     [4] 

396.  Line  A(-  6",  -  1",  -  I'M  B(-  3",  +  V',  0")  and  line  parallel  to  AB  through 
point  C(-4",-r,  +  r0. 

397.  Lines  through  points  D(-  6",  -  1",  +  1")  and  E(-  4",  +  1^'',  +  fO,  parallel 
to  H  and  making  45 '^  with  Y. 

398.  Lines  parallel  to  the  ground  line,  through  points  G(- 5",  -  J",  -  1^")  and 

H(-4",-i-,-r). 

APPLICATIONS. 

399.  [4]  The  Hne  A(- 600',  +  50',  +  25')  B(- 340',  +  200',  +  190')  is  a  tele- 
phone wire  running  from  a  valley  town  to  a  camp  on  the  side  of  a  mountain.  A 
house  on  the  mountain  at  the  point  H(- 280',  +  70',  +  150')  is  to  be  connected  to 
the  line  AB  by  the  shortest  possible  wire.  Find  the  projections  and  length  of  the 
wire.    Scale,  1"  =  100'. 

400.  [4]  For  supporting  a  boom  derrick  a  wire  cable  is  to  be  run  from  a  point 
M(-  130',  +  100',  +  90')  on  its  mast  to  a  point  X  in  a  beam  A(-  260',  +  100',  +  55') 
B(- 160',  +  15',  +  55').  Find  the  projections  and  length  of  the  shortest  possible 
cable  that  can  be  used.    Scale,  1"  =  40'. 

401.  [4]  In  the  cottage  roof  of  Fig.  19,  find  the  distance  from  the  point  N  to 
the  hip  rafter  AC. 

402.  [4]  In  the  cottage  roof  of  Fig.  19,  find  the  distance  from  the  point  C  to 
the  hip  rafter  AE. 

403.  [2]  In  the  ventilator  cap  of  Fig.  20,  find  the  distance  from  the  corner  O 
to  the  lines  KM  and  MN. 

404.  [2]  In  the  derrick  of  Fig.  27,  find  the  distance  from  the  mast  top  D  to 
the  boom,  also  show  the  projections  of  a  wire  running  from  the  post  C  perpendic- 
ular to  the  guy-wire  DB. 


23 

405.  [2]  In  the  skew  bridge  of  Fig.  28,  find  the  distance  from  the  point  M  to 
the  diagonal  DL ;  also  from  the  point  M  to  the  portal  post  AC,  and  from  the 
point  C  to  the  portal  post  BD.    Use  scale,  1''  =  10'. 

406.  [4]  The  line  A(- 5',  -  2',  -  i')  B(- 3-^,  -  i',  -  U')  is  the  center  line  of 
a  gas  pipe.  A  pipe  for  a  light  at  the  point  L(-  H',  -  f ',  -  If)  is  to  connect  with 
the  first  pipe  by  a  right  angled  elbow.  Find  the  length  of  the  latter,  making  no 
allowance  for  the  elbows.     Scale,  1"  =  1'. 

407.  [4]  A  soldier  at  target  practice  fired  in  the  line  A(- 6",  +  2",  + 1") 
B(-3",-f  1'',  +  !").  By  how  much  did  he  miss  the  center  0(- Sf,  +  1",  +  y') 
of  a  target? 

408.  [4]  A  mountain  railroad  takes  the  direction  M(- 1|", +  ^",  0") 
N(-  7",  +  2Y',  +  lY').  A  hut  on  the  same  slope  at  the  point  H(-  5^",  +  f",  +  Y') 
is  how  far  from  the  railroad  ?    I<et  V  =  ^  mile. 

409.  [4]  A  sewer  pipe  has  the  line  B(- 37',  +  18',  +  7')  C(- 25',  +  3',  +  4')  as 
a  center  line.  What  is  the  shortest  waste  pipe  that  could  be  put  in  from  a  drain 
at  the  point  M(-  40',  +  6',  +  16")  ?    Scale,  ^"  =  I'-O". 

410.  [4]  Theline  A(-6",  +  2",  +  r)  B(- Sf',  +  T,  +  3")  is  the  center  line  of 
a  diagonal  member  of  a  roof  truss,  to  be  connected  to  point  M(-  3|",  +  1^",  +  2") 
by  a  member  perpendicular  to  the  first.     Find  its  length. 

411.  [4]  The  line  A(- 93',  +  45',  +  15')  B(- GO',  +  10',  +  15')  is  the  center 
line  of  a  mine  shaft.  It  is  proposed  to  connect  with  this  shaft  by  a  tunnel  from  a 
point  M(-  100',  +  15',  +  5')  in  another  shaft.  Find  the  length  of  the  shortest  pos- 
sible tunnel.    Scale,  1"  =  20'. 

412.  [4]  A  stroke  of  lightning  was  estimated  to  have  taken  the  line 
A(-60',  +  3',  +  25')  B(-35',  +  16',  +  2').  By  how  much  was  an  ornament 
0(-42', +  5',-^20')  on  a  church  spire  missed?    Scale,  1"  =  10'. 

13.— TO  FIND  THE  ANGLE  WHICH  A  GIVEN  RIGHT  LINE  MAKES 

WITH  A  GIVEN  PLANE. 
General.    [4] 

413.  Line  A(- 5",  -  V',  0")   B(- 3",  -  1^",  -  2"),  plane  R(- 2",  +  150°,  -  150°). 

414.  Line  C(- 6",- l",-!")   D(- 4",- l",-2"),  plane  S(- 3",+ 112^°,- 150°). 

415.  Line  E(- 4^",+ U"  -  1")    F(- 3",  4  i",  +  1"),  plane   T(- 6",- 30°,  -  60°). 

When  plane  is  perpendicular  to  H  or  V  or  both.    [4] 

416.  Line  A(- 4", +  4",  0")    B(- 2^",  +  2",  +  1^"),  plane  R(- 6",  +  30°,  -  90°). 

417.  Line  C(- 6",  0",  +  1")   D(- 3",  -  H",  -  H"),  plane  S(- 3|",  +  90°,  +  45°). 

418.  P  at  -  3".     Line  E(-  3",  +  1",  -  i")  F(-  1",  -  |",  -  1")  and  plane  P. 

When  plane  is  parallel  to  ground  line.    [4]    P  at  -  3". 

419.  Line  A(- 3",  -  i",  -  2")    B(- 1",- i", -i")    and  plane  R(  oo,  -  1",  -  2"). 

420.  Line  C(- 3",  +  If,  -  1|")   D(- 2",  +  1",  -  J"),  plane  S(  oo,  -  H",  -  f ). 

421.  Line  E(-3",-2",-U")  F(-f,-i",  +  f)  and  plane  T(  oo,  -  If,  oo). 

APPLICATIONS. 

422.  [2]  P  at  -  3".  An  inclined  guide  pulley  shaft  has  center  line 
A(-2',-2f',-3J:')  B(-4',-^',-l^').  What  angle  does  it  make  with  the  floor, 
(H  plane)  the  front  wall  (the  V  plane)  and  with  a  roof  plane  R(  co,  -  2f',  -  If). 
Scale,  1"  =  1'. 


24 

123.  [4]  Rays  of  light  in  the  direction  A(-  6'',  +  Y',  +  f")  B(-  3i'',+  f",+  2^") 
are  reflected  from  a  mirror  plane  M(- IfJ",  +  45°,  -  30°).  What  is  the  angle  of 
incidence  ? 

424.  [4]  A  vertical  telephone  pole  stands  on  a  hillside  S(- 6^",  -  60°, +  45°). 
What  angle  does  the  pole  make  with  the  plane  of  the  hill? 

425.  [2]  The  line  A(- Gf,  +  2^,  -  i')  B(- 1-^',  +  4',  +  2f')  is  the  center  line 
of  a  tie  rod  in  a  roof  truss.  What  angle  does  the  tie  rod  make  with  the  floor 
plane  (H  plane)  and  what  angle  with  the  roof  plane  determined  by  the  members 
M(-7^',f5',  +  lf')  N(-4r,  +  ir,  +  2f')  and  NP(- 1|',  +  2f ,  -  1^')-  Scale, 
1"  =  1'  0". 

42f).  [2]  In  the  roof  of  Fig.  19.  find  the  angle  which  the  valley  rafter  MN 
makes  with  the  roof  plane  U. 

427.  [2]  In  the  roof  of  Fig.  19,  find  the  angle  which  the  ridge  AB  makes  with 
the  roof  plane  T. 

428.  [2]  In  the  skew  bridge  of  Fig.  28,  find  the  angle  that  the  portal  post  DB 
makes  with  the  plane  of  the  diagonals  DL  and  DM  ;  also  the  angle  which  the  por- 
tal brace  EI  makes  with  the  plane  of  CD  and  CN.    Use  scale,  1"  =  10'. 

429.  [2]  In  the  roof  of  Fig.  29,  find  (1)  the  angle  which  the  ridge  8-14  makes 
with  the  roof  plane  S,  (2)  the  angle  which  the  same  ridge  makes  with  the  plane 
X.    Use  scale,  Y'  =  I'-O''. 

430.  [4]  In  Fig.  31,  find  the  angle  which  the  rope  AB  makes  with  the  south 
bank  plane  S. 

431.  [2]  Find  the  angle  which  the  telegraph  wire  PW  of  Fig.  30  makes  with 
the  walls  A  E  F  B,  F  B  C  G,  the  left  end  wall  of  the  factory,  and  the  roof. 

432.  [4]   What  angle  does  the  tow-rope  of  Fig.  32  make  with  the  river  bank  S? 

433.  [4]  What  angle  does  the  tow-rope  of  Fig.  32  make  with  the  floor  of  the 
canal  boat? 

434.  [2]  In  the  reducer  of  Fig.  24,  what  angle  does  the  edge  between  planes 
A  and  B  make  with  the  plane  C?    Use  scale,  1"  =  V  -0". 

435.  [2]  Fig.  27  shows  a  Boom  Derrick.  Find  (1)  the  angle  between  the 
boom  FE  and  the  plane  determined  by  the  guy  wires  BD  and  CD,  (2)  the  angle 
between  the  guy-rope  BD  and  the  plane  determined  by  the  points  A,  B  and  C. 

14.— TO  FIND  THE  ANGLE  BETWEEN  TWO  GIVEN  PLANES. 

General.    [4] 
43G.     Planes  P(-6", -45°.  +  G7^°)   and  Q(-3",  +  60°,-45°). 

437.  Planes  R(- 6'',  +  67^°,  f  120°)  and  S(- 3",  +  135°,  + 60°). 

438.  Planes  T(-6'',  +  67r,-30°)  and  U(- 2",  +  112|°,  -  120°). 

Including  planes  perpendicular  to  H  or  V,  or  both.    [4] 

439.  Planes  Pf- 6",  4- 90°, -30°)  and  Q(- 3^", +  135°, -135°). 

440.  Planes  R(- 5".  H  90°, -30°)  and  S(- 3", +  135°, +  90°). 

441.  Planes  T(-r/',  + 30°,- ■?2^°)  and  U(- 3'',  -  90°,  +  90°). 

Including  planes  parallel  to  ground  line.    [4] 

442.  Plane  P(- 6", -45°, +  45°)  and  Q(  00,  + 2",^  1"). 

443.  Plane  R(  x,  -  1^",  -  2")  and  S(  oo,  +  1^",  -  1"). 

444.  Plane  T(- 6'', +  45°-, -30°)   and  U(  oo,  -  1'',  co). 

When  one  of  planes  is  a  plane  of  projection.    [4] 

445.  Plane  R(- 6", +  45°, -30°)  and  H. 

446.  Plane  S(-6",  +  67r,-60°)  and  V. 

447.  Plane  TC- 2", +  90°, -30°)  and  P  at -4". 


25 

ONlv  TRACE  OF  A  PLANE  BEING  GIVEN,   AND  THE  ANGLE  WHICH   THIS  PLANE   MAKES 
WITH  A  PLANE  or  PROJECTION,  TO  EIND  THE  OTHER  TRACE.     [4] 

4-18.     The  H  trace  of  a  plane  R  is  given  as  (-6'', +  30°).     The  plane  makes  an 
angle  of  15°  with  H.     Find  the  Y  trace. 

449.  The  V  trace  of  a  plane  S  is  given  as  (-(>",  +  45°).     The  plane  makes  an 
angle  of  60°  with  H.    Find  the  H  trace. 

450.  The  H  trace  of  a  plane  T  is  given  as  ( 20,  -  1|")-    The  plane  makes  an  an- 
gle of  30°  with  H.     Find  the  V  trace. 


b' 


APPLICATIONS. 

451.  [8]  In  the  cottage  of  Fig.  19,  find  the  bevel  angle  of  the  hip  rafter  AC, 
that  is,  the  angle  between  the  planes  T  and  R.  Also  find  the  valley  angle  between 
the  planes  R  and  V. 

452.  [2]  In  the  cottage  of  Fig.  19,  find  the  bevel  angle  of  the  hip  rafter  AH, 
that  is,  the  angle  between  the  roof  planes  T  and  U.  Also  find  the  bevel  angle  for 
the  ridge  rafter  AB. 

453.  [2]  In  the  skew  bridge  of  Fig.  29,  find  the  angle  between  the  portal  plane 
AP)DC  and  the  bridge  floor.  Also  the  angle  between  the  portal  plane  and  the 
plane  of  one  of  the  trusses  of  the  bridge.    Use  scale,  1"  =  10'. 

454.  [2]  In  the  sheet  metal  reducer  shown  in  Fig.  24,  find  the  plane  angles  for 
corner  angle-irons,  that  is,  the  angles  between  the  planes  A  and  B,  B  and  C,  etc. 
Use  scale,  1"  =  V  -  0''. 

455.  [2]  In  the  house  of  Fig.  29  find  the  bevel  angle  for  the  hip  rafter  1-5,  that 
is,  the  angle  between  the  planes  R  and  S.  Also  find  the  valley  angle  between 
S  and  T,  and  the  bevel  angle  for  hip  rafter  between  planes  T  and  X,  Use  scale, 
J/'  =  I'-O''. 

456.  [2]  In  the  house  of  Fig.  29,  find  the  bevel  angle  for  the  hip  rafter  11-14, 
that  is,  the  angle  between  planes  X  and  W.  Find  also  the  bevel  angle  for  the 
ridge  rafter  5-6,  and  the  vallev  angle  between  planes  S  and  T.  Use  scale, 
Y'  =r  -  0". 

457.  [2]  In  the  ventilator  cap  shown  in  Fig.  20,  find  the  angles  between  the 
planes  D  and  B,  between  A  and  B,  between  A  and  the  plane  of  the  cap  base, 
between  B  and  the  plane  of  the  cap  base, 

458.  [2]  In  the  gondola  car  body  of  Fig.  23,  find  the  angles  between  plates 
C  and  D,  between  A  and  C,  between  C  and  the  door  plate. 

459.  [4]  The  line  A(- G'',  +  1",  +  f")  B(- 4^",  +  2-|'',  +  2^")  and  a  line  par- 
allel to  AB  through  the  point  C(- 4f",  +  J",  +  1:J")  are  the  back  edges  of  a  chan- 
nel iron  connected  by  a  bent  plate  to  a  plane  T(-  3if' ,  -  150°,  +  120°).  Find  the 
flare  angle  for  the  bent  plate.     See  Fig.  41. 

460.  [4]  A(  -6",  +  U'',-f  If")  B(-4r,  +  lF',  +  r')  and  a  line  parallel  to  AB 
through  C(- 4f",  +  2",  +  l:f")  determine  the  web  plane  of  an  I  beam  which  is 
riveted  by  a  bent  plate  connection  to  a  horizontal  bed  plate.  Find  the  flare  angle 
for  the  bent  plate.     See  Fig.  40. 

461.  [2]  The  top  edges  of  the  slope  planes  of  a  dry  dock  form  a  rectangle 
100'  by  60'.  These  planes  slope  toward  the  center  at  an  angle  of  67^°  with  the 
horizontal.  Find  the  traces  of  the  planes,  their  intersection  with  each  other,  and 
their  intersection  with  the  bottom  of  the  dry  dock,  40'  below  the  top  rectangle. 
Scale,  1"  =  30'. 

462.  [2]  The  line  A(- 70',  +  35',  0')  B(- 37',  ^  5',  0')  is  the  H  trace  of  a  rail- 
road embankment  plane  whose  batir  is  1  horizontal  to  2  vertical,  and  whose  ver- 
tical height  is  20  feet.     Find  the  top  line  of  the  slope  plane,  the  intersection  of 


26 

this  plane  with  a  hillside  H(- 48',  -  90°,  +  30°),  and  the  angle  between  the  em- 
bankment plane  and  the  hillside.     Scale,  1''  =  10'. 

463.  [3]  The  line  M(- 50',  +  9',  0')  N(- 17',  +  40',  0')  is  the  trace  on  the 
ground  of  a  wing  wall  slope  plane  whose  batir  is  1  horizontal  to  2  vertical.  The 
top  line  of  the  wall  is  parallel  to  the  ground  trace  and  25  feet  above  the  ground. 
Find  this  top  line,  the  intersection  of  the  wing  wall  plane  with  the  embankment 
plane  T(  cc.  +  30',  -h  15')  and  the  angle  between  wing  wall  and  embankment  plane. 
Scale,  1"  =  10'. 

464.  [2]  A  tower  roof  is  to  be  built  in  the  same  general  shape  and  position  as 
shown  in  Fig.  20,  for  the  ventilator  cap.  The  planes  D  and  B  are  to  have  a  slope 
of  1-^  vertical  to  1  horizontal,  while  the  planes  C  and  A  are  to  have  a  slope  of  2 
to  1.  Find  the  projections  of  the  roof  and  the  angles  between  the  planes  C  and 
D,  and  between  planes  D  and  B. 

15.— TO  FIND  THE  SHORTEST  DISTANCE,   XY,   BETWEEN  TWO 
RIGHT  LINES  NOT  IN  THE  SAME  PLANE. 

General.    [2] 

465.  Lines    A(- 6",  f  2",  +  2")    B(- 3",  -  1^",  +  If )    and    C(- 5",  +  If ,  +  |") 
D(-3",ff",-F2i"). 

466.  Lines    E(- 5J-", -"  2^',  + 1")    F(- 2f ,  +  1",  -  2")    and    G(- 5^",  -  3",  0") 
H(-  2V,  0'',  -^2^"). 

467.  Lines  'l(-  6'',  -  f',  -  P/')   J(-  U/',  -f  V/',  -  D   and   K(-  3^,  -  U",  -  D 
L,(_  3^"^  _  1''^  -1"). 

468.  Lines  "m(- 6'', -^2^,-1-1")    N(- 3f , -!- -f,  +  1")    and   0(- 6",  +  f ,  +  2f  ) 
P(-  3J.",  +  2",  +  f ). 

469.  Lines    Q(- :r,  + 2",  1-41")    R(- 5f ,  + 3|",  + 3|"),    S(-'2|",  +  2i",  +  2") 

T(-r>+r,+'ir)- 

Including  lines  parallel  to  the  planes  of  projection.    [2] 

470.  Lines    A(- 51",  -  T,  -  D    B(- 5^',  -  T,  -  U")    and    C(-4",  +  H,-r) 
D(-3",-|",-l"). 

471.  Lines  E(- 6",  -  2",  -  2")    F(- 5r,  -  T,  "  21")    and  G(- 3^",  -  H",  -  D 
H(-3r,-in-3"). 

472.  Lines  J(-  6^",  -  2",  -  2")   K(-  2^",  +  2^",  +  £")   and  L(-  4f",  +  If",  +  i") 
M(-U",  +  i-2",  ^4i"). 

APPLICATIONS. 

473.  [1]  The  line  C(- 23',  +  2',  +  4')  D(- 15',  h- 7',  +  4')  is  the  center  line  of 
a  telegraph  wire.  It  is  proposed  to  cross  this  line  with  a  high  tension  transmission 
line  whose  center  line  is  A(- 23',  +  8',  + 10')  B(- 11',  +  2',  + 2').  If  the  safe  dis- 
tance between  such  wires  is  H  feet,  is  this  crossing  safe  or  not,  and  by  how 
much?    Scale,  y'  =  l'-0". 

474.  [2]  The  center  lines  of  two  gas  pipes  which  are  to  be  joined  by  means 
of  one  straight  length  of  pipe  and  two  right  angled  elbows  are  A(- 6',  -  2',  -  2') 
B(-5i',-J',-2i')  and  M(- 3^',  -  U',  -  f)  N(- 3^',  -  U',  -  3').  Find  the 
shortest  possible  connection  of  this  sort  which  can  be  made,  allowing  1^"  at  each 
end  for  the  elbow.     Scale,  1"  =  1'  -  0". 

475.  [2]  The  line  L(- 9^',  -  1',  -  3^)  M(- 3',  -  9.',  -  3^)  is  the  center  line  of 
the  main  shafting  in  a  certain  shop.  It  is  proposed  to  run  an  electric  light  wire  in 
the  direction  J(-  13',  +  4',  4  4')  K(-  5',  -  1|',  -  4|').  Find  the  distance  between 
the  two  at  their  closest  point.    Scale,  |"  =  I'-O". 


27 

GENERAL  PROBLEMS  BASED  ON  POINT,  LINE  AND  PLANE, 
WITH  APPLICATIONS. 

480.  [2]  A  2V^  cube  stands  upon  H  in  the  first  quadrant,  with  one  of  its  faces 
making  22^°  with  V.  Find  (1)  the  H  and  V  projections  of  the  cube,  (2)  the 
true  length  of  a  diagonal,  (3)  the  distance  from  one  corner  to  the  plane  of  the 
three  adjacent  corners. 

481.  [2]  A  horizontal  trapeze  bar  21  feet  long  is  hung  by  vertical  ropes  4^ 
feet  long  attached  to  its  ends.  How  far  would  this  bar  be  raised  by  turning  it 
through^)0°  ?    Use  scale,  !''  =  !'-  0". 

482.  [1]  Locate  P  at  -54",  and  ground  line  parallel  to  shorter  edges  of  sheet. 
The  point  M(- f", -!-|",  +  1")  is  revolved  about  the  line  A(+ |-",  +  3^'', +  4i") 
B(+3'',  +  1",  0").  Find  (1)  the  points  where  this  point  pierces  H,  V  and  P, 
(2)  the  H,  Y  and  P  projections  of  the  point  M  after  180°  revolution  from  its 
original  position. 

483.  [1]  P  at  -  T".  Scale,  i''  =  1'  -  0".  Revolve  the  point  A(-  5V,  ~  If,  -  l^') 
about  the  line  E(-  41',  -  5',  -  o|')  C(-  V,  -  V,  -V)  as  an  axis  and  find  the  points 
where  it  pierces  H,  V  and  P  in  this  revolution. 

484.  [1]  P  at  -7V\  Scale,  l"  =  l'-0''.  Let  H,  V  and  P  represent  respec- 
tively the  ceiling  and  end  and  side  partitons  of  a  factory  room.  The  point  x\  on 
the  rim  of  a  guide  pulley  revolves  about  the  line  BC  which  is  the  center  line  of 
the  pulley  shaft.  Find  the  points  where  this  point  A  pierces  H,  V  and  P  in  the 
revolution.     Also  find  the  projection  of  A  after  30°  revolution. 

A(-  4'5",  -  l'(r,  -  1'21")    n(~  3'1",  -  4'0",  -  4'24-")    C(-  O'Ci",  -  PO",  -  0'5"). 

485.  [1]  (Ground  line  parallel  to  shorter  edges  and  If  below  middle  of  sheet). 
P  at-5|".  Scale,  f  =  I'-O''.  The  three  planes  H,  V  and  P  represent  respec- 
tively the  floor,  and  two  partitions  of  a  power  house.  The  point  X,  4'-0'''  above  the 
floor,  4'-0"  in  front  of  V,  and  2'-  5V'  to  the  right  of  P,  is  a  point  on  the  circum- 
ference of  a  guide-sheave  for  a  wire  rope  drive.  The  center  line  of  the  axis  of 
the  wheel  pierces  H  6''  in  front  of  V  and  4'-5''  to  the  right  of  P  and  pierces  V 
I'-O''  below  H  and  5'-10i"  to  the  right  of  P.  Find  the  points  where  the  point  X 
cuts  through  the  floor  and  partitions  as  it  revolves  about  the  axis.  Also  find  the 
projections  of  the  point  X  when  it  has  revolved  through  an  angle  of  60°. 

48G.  [1]  Scale,  J/' =  I'-O".  The  line  A(- 48',  +  16',  +  8')  B(- 28',  +  2'.  +  8') 
is  the  axis  of  the  shaft  of  a  fly  wheel  in  a  power  station  whose  floor  and  wall  are 
represented  by  H  and  V.  The  point  0(- 30',  h  11',  +  18')  is  in  the  center  line  of 
the  rim  of  a  fly  wheel  to  be  located  upon  the  shaft.  The  width  of  the  fly  vv^heel 
face  parallel  to  the  shaft  axis  is  to  be  2',  that  is,  1'  on  each  side  of  O.  Find 
(1)  the  radius  and  projections  of  the  proposed  fly  wheel,  (2)  the  holes  which 
must  be  cut  from  the  wall  and  floor  to  allow  a  clearance  of  6"  all  round  the  fly 
wheel  for  safety. 

487.  [2]  Given  the  line  M(- 6",  +  1",  +  i")  N(- 3",  +  3",  +  2")  and  the  line 
A(-7",  +  2y',  +  2i")  B(-4",  +  4",-U").  Find  the  projections  of  AB  after  it 
has  been  revolved  about  MN  as  an  axis  through  an  angle  of  75°. 

488.  [1]  Pass  a  sphere  through  the  four  points  B(- 10",  +  li",  - 1") 
D(-8f  ,  +  3",  +  2J:")  G(-7i-",+  li",  0")  and  W(-7|",  +  r',-f"). 

489.  [2]  Circumscribe  a  sphere  about  the  triangular  pyramid  whose  vertices 
are  the  points  A(- 5",  +  24",  +  2")    B(- 6§",  +  3|",  0")   C(- 3§",  +  3f",  0")   and 

D(-4r,+ino"). 

490.  [2]  The  following  four  points  M(- 5", -f  24",  +  2")  N(- 6f".  +  3|",  0") 
O(-3|",  +  3§",0")  P(_4i",  +  li",  0")  are  the  vertices  of  a  tetahedron.  Find 
the  inscribed  sphere. 


28  GENKRAl,   PROBLEMS    RASED   ON    POINT,    UNE   AND  PLANE 

491.  [2]  The  points  A(- 7",  -  3",  0")  B(-5",-r,0")  C(- 2^'',  -  2^^  O'O 
and  D(-4^'',  -  If",  -  2|")  are  the  vertices  of  a  tetrahedron.  Find  the  inscribed 
sphere. 

492.  [2]  Let  H,  V  and  T(- 2",  -  120°,  +  135°)  represent  respectively  the  floor, 
the  wall,  and  the  roof  plane  in  a  corner  of  a  garret.  A  perfectly  elastic  ball  is 
thrown  in  the  direction  of  the  line  A(- 7^",  +  2",  +  2")  B(- 5",  +  2",  +  2"), 
strikes  the  roof  plane  T,  bounds  off  and  strikes  the  floor  H,  then  bounds  off  of  H 
and  strikes  the  wall  Y.    Find  the  3  points  X,  Y  and  Z  where  it  strikes  these  planes. 

493.  [2]  The  plane  T(- 6|",  -  45°,  +  60°)  and  the  planes  of  projection  H  and 
V  are  three  mirrors,  from  which  a  ray  of  light  A(- 2^",  +  2-|",  +  2-') 
B(- 4^",  f  2|'',  h-J")  is  successively  reflected  at  the  points  X,  Y  and  Z.  Find 
these  points. 

494.  [2]  P  at  -  2".  Construct  the  H  and  V  projections  of  a  cube,  in  the  first 
quadrant,  one  of  whose  faces  is  in  the  plane  F(- 3|",  -  30°,  +  60°)  with  the  side 

A(-2r,+r,z)  B(-r,+ir,z). 

495.  [2]  Find  the  H  and  V  projections  of  a  regular  square  pyramid  whose  ver- 
tex is  at  the  point  0(- 3",  -  3^",  -  2^")  and  whose  1|"  square  base  is  in  the 
plane  T(- 2i",  +  150°,  -  120°). 

496.  [1]  Find  the  projections  of  a  regular  hexagonal  pyramid  having  its  base 
in  a  plane  through  A(-  10:^",  +  If,  +  2-f ')  and  par.  to  lines  B(-  12'',  +  ^'%  +  4") 

C(-6i",  +  r',  +  3r)  and  D(-9r,  +  r.  +  'ir)  E(- 6^, +  4", +  ^'0-  One  side 
of  the  base  lies  in  the  line  of  intersection  of  the  plane  of  the  base  and  a  plane 
through    A    perpendicular    to    line    BC.      Vertex    of    the    pvramid    at    point 

0(-5",  +  4i'',  +  5^'')- 

497.  [2]  Construct  a  line  XY,  parallel  to  the  plane  T(- 6^',  -  30°,  +  60°)  and 
1'' therefrom,  which  cuts  the  lines  M(- 6'",+ 2f ,+ 2f")  N(- 4^",+ 1",  f  |'')  and 

o(-4'',+ir,+2D  p(-ir",+ir,+D- 

498.  [2]  Given  the  points  A(-6",  +  l",  0")  B(-3a",  +  f,  0")  and 
C(-2f , +  3|'',  0")  as  the  vertices  in  H  of  the  base  of  a  triangular  pyramid, 
whose  vertex  is  at  a  point  F,  such  that  the  lateral  edges  AF,  BF,  and  CF  are 
4^',  4f  and  4"  in  length  respectively.  Find  the  H  and  V  projections  of  the 
pyramid. 

'499.  [2]  Scale,  1"  =  50'.  Vertical  drill  holes  at  points  A(- 260', +  60',  + 180') 
B(- 310',+ 175',+  110')  and  C(- 200',  +  55',  +  110')  of  a  hillside  strike  an  ore 
vein  at  depths  of  170',  210'  and  70'  respectively.  Find  the  line  of  outcrop  of  the 
vein,  that  is,  the  intersection  of  the  ore  plane  with  the  plane  of  the  hillside. 

500.  [2]   Find  a  line  XY  which  touches  the  three  lines  A(- 6", -i",  -  2^")' 

B(-5",-2",-i"),c(-4r,-3",-2r)  D(-3r,-r,-r),andE(-3",-ir,-r) 

F(-l",-3",-3"). 

501.  [2]  through  the  point  0(-  4|",  -  1^",  -1^")  pass  a  line  parallel  to  planes 
T(-6i",  +  30°,-50°)  and  S(- 1",  +  115°,  -  140°). 

502.  [4]  Pat -I".  The  vertical  line  A(- 51",  + li",  + 2")  B(- 5^",  + li",0") 
is  a  body  diagonal  of  a  cube  in  the  first  quadrant  with  its  lowest  corner  on  H  at 
the  point  B.    Find  the  three  projections  of  such  a  cube. 

503.  [2]  Through  the  point  M(- 6|",  - 1",  -  lJ:")construct  the  line  MXY 
touching  the  lines  P(- 6",  -  2", -3")  Q(- 5",  -  i",  -  ^")  and  R(- 4",  -  |",  -  3") 
S(-2",-3",-f"). 

504.  [2]  Through  the  point  GC-  5f",- 1",-  1|")  construct  a  line  GXY  touching 
the  lines  M(- 4^",  -  3",  -  2^")  N(- 3^",  -  i",  -  i")  and  0(- 3",  -  1^", -i") 
P(-l", -3",-3"). 

505.  [2]  Through  the  line  M(- 4f",  -  2|",  z)  N(-2",-^",  z)  in  the  plane 
R(-7i",  +  45°,-30°)     pass  a  plane  T  making  an  angle  of  30°  with  R. 


GENERAI,   TROBLTvAIS   BASED   ON    POINT,    LINE   AND   PLANE  29 

506.  [2]  Through  the  Hne  0(-6",  y,-V')  P(-2r',  y,  +  H")  in  the  plane 
S(-tV,  -  150°,  +  130°)  pass  a  plane  U  making  an  angle  of  45°  with  S. 

507.  [2]  Find  line  CD,  parallel  to  A(-6r,-r,-r)  B(- of',  -  T,  -  ^D 
and  intersecting  the  line  M(- 6.^,  -  2'',  -  3")  N(- 5i", -i", -^'0  and  the 
line  P(-  4r,  -  r,  -  2r)  Q(-  H'',  "  3",  -T )• 

508.  [2]  Find  a  plane  R  which  bisects  the  dihedral  angle  between  planes 
S(- 11'', -60°, +  30°)   and  T(- T-f, +  -iO°,  -  45°).  . 

509.  [2]  Find  the  point  K  in  line  A{- 1^',  +  !^,  + H")  ^(- H'',  + V',  +  D 
which  is  equidistant  from  planes  S(- 7'',  -  60°,  +  30°)  and  T(-f'', +  45°,  -  60"°). 

510.  [2]Through  point  C)(- 4'^  +  3",  +  2|")  pass  a  plane  T  making  an  angle 
of  60°  with  H  and  67-^°  with  V. 

511.  [1]  Construct  the  projections  of  four  spheres  of  radii  1^",  2",  2^"  and  1" 
respectively,  so  located  that  each  sphere  is  tangent  to  the  other  three. 

512.  [2]  Given  two  points  0(- H",  -  ^i",  -  D  and  K(- 2r,  -  i",  -  2D 
and  a  mirror  plane  M(-  7^",  f  30°,  -  45°).  Find  the  projections  of  a  ray  of  light 
which  emanates  from  O  and  after  being  reflected  from  M  passes  through  the 
point  K. 

513.  [2]  If  the  direction  of  rays  of  light  is  given  parallel  to  the  line 
I/_6i", +  2'', +  2^")  M(-4", +  1'', +  |'0  find  the  shadow  cast  upon  the  hori- 
zontal plane  of  projection  by  a  certain  triangle  A(- 6|",  +  4-|-",  +  f") 
B(-  4'^  +  2r,  +  3^0   C(-  4^',  +  5r,  +  1"). 

514.  [2]  A  ray  of  light  passes  through  the  points  A(-iy',  +  4^",  +  31")  and 
B(-  33/',  +  If',  +  2")  and  is  reflected  from  a  mirror  M(-  |""+  60°,  -  45°).  Find 
the  projections  of  the  incident  and  reflected  rays,  and  the  angle  of  incidence. 

515.  [1]  P  at  -8".  The  top  of  an  inclined  draughting  table  is  a  rectangle, 
with  three  corners  at  the  points  A(- 7^",  -  1",  -  4f")  B(- 7|^  -  5'', -3^") 
C(-2'',-5", -3rO-  An  electric  light  at  the  point  L(- 4f ',  -  3f',  -  1'')  is 
guaranteed  to  light  satisfactorily  at  this  distance  within  a  cone  of  maximum 
intensity  rays,  whose  elements  make  an  angle  of  30°  with  the  vertical.  Find  the 
three  projections  of  sixteen  of  these  rays  and  then  approximately  the  curvf 
bounding  that  portion  of  the  table  within  the  cone  of  maximum  intensity. 

516.  [2]  Two  pulleys  are  located  with  their  pitch  circles  in  the  same  plane 
T(- 12',  -  45'^,  +  60°)  and  are  2'  and  3'  in  diameter  respectively.  The  face  01 
each  pulley  is  12",  their  shafts  are  6'  apart,  and  they  are  connected  by  an  open  belt 
9"  wide.  Find  projections  of  pulleys  and  belt,  and  length  of  the  latter,  making 
no  allowance  for  slack.     vScale,  Y'  =  I'-O". 

517.  -[1]  Scale,  J"  =  I'-O''.  Two  pulleys  of  3'  diameter  are  located  at  points 
A  of  shaft  A(-  24'",  -  6',  -  IV)  B(-  24',  -  6',  -  6')  and  C  of  C(-  8',  -  IV,  -  7^') 
D(- 8',  -  6',  -  7|').  They  are  to  be  connected  by  a  belt  running  over  two  guide 
pulleys  of  3'diameter,  so  located  that  the  belt  will  be  as  short  as  possible  and  will 
run  in  either  direction.  Find  its  length  and  center  line  projections,  also  the  pitch 
line  projections  of  the  guide  pulleys.  (Hint.  A  belt  must  always  be  delivered  in 
the  plane  of  the  pulley  toward  which  it  is  running.    .See  Fig.  33.) 

518.  [1]  G.  L.  par.  to  short  edges  of  sheet.  Scale,  4"  =  I'-O".  Assuming  the 
point  D  as  per  dimensions  in  Fig.  33,  find  the  projections  of  the  main  and  guide 
pulleys  and  belt,  assuming  all  pulley  faces  to  be  9",  width  of  belt  6",  shaft  diame- 
ters 4".     (See  hint  in  problem  above.) 

519.  [1]  G.  L.  par.  to  short  edges  of  sheet.  Scale,  4"  =  I'-O".  In  Fig.  33, 
locate  a  guide  pulley  as  indicated,  but  in  such  a  position  as  to  give  the  shortest 
possible  belt.  Show  only  pitch  line  of  guide  pulley  and  of  belt.  Find  length 
of  belt. 


30  GENERAL  PROBLEMS   BASED  ON   POINT^   LINE  AND  PLANE 

520.  [1]  Scale,  |''  =  I'-O".  Find  the  projections  of  two  pulleys,  diameters  3' 
and  4',  to  be  located  at  points  A  and  B  respectively  of  shafts  A(-  25',  -  4^',  -  3') 
C(-32',-U',-l')  and  B(- 11^,  -  n',  -  4')  D(- 13^,  -  5^',  -  i'),  and  to  be 
connected  by  a  belt  running  over  two  guide  pulleys  of  2^'  diameter,  so  located 
that  the  belt  will  be  the  shortest  possible  and  run  in  either  direction.  (Hint:  A 
belt  must  always  be  delivered  in  the  plane  of  the  pulley  toward  which  it  is  run- 
ning.) 

521.  [1]  On  a  certain  set  of  plans,  the  projections  of  a  ventilator  cap  are  as 
shown  in  Fig.  20.  Find  (a)  the  angle  between  planes  D  and  B,  that  is,  the  flare 
angle  for  the  ridge  angle-iron,  (b)  the  angle  between  planes  A  and  B,  that  is,  the 
flare  angle  for  the  hip  angle  irons,  (c)  the  angle  which  the  ridge  makes  with  the 
plane  A,  (d)  the  angle  which  the  planes  D  and  C  make  with  a  horizontal  plane 
through  the  base  of  the  cap,  (e)  the  pattern  (true  size  and  shape)  for  the  cap, 
allowing  3"  for  lap  where  necessary. 

522.  [2]  Fig.  21  represents  in  outline  the  projections  of  a  lamp  shade.  Find 
the  following  values,  which  would  be  used  in  its  construction:  (1)  Angle  be- 
tween planes  A  and  B,  (2)  angle  which  planes  A  and  B  make  with  a  horizontal 
plane  through  the  base  of  the  shade,  (3)  angle  which  planes  A  and  B  make  with 
a  vertical  plane  through  the  edge  FG,  (4)  angle  which  edges,  as  EF,  make  with 
the  top  T,  (5)  angle  which  edge  EF  makes  with  the  base  edge  FG,  (6)  size  of 
glass  planes,  allowing  I''  inside  of  extreme  edges  as  EF  and  FG. 

523.  [1]  Scale,  ^"  =  I'-O".  The  sketch  in  Fig.  22  is  a  steel  tank  for  an  elevator 
boot,  the  plates  to  be  bent  and  riveted  as  shown.  Dimensions  of  open  top  are 
5'_0''  by  3'-0",  of  bottom  plate  without  laps  3'-0"  by  18".  Height  of  boot  is 
2'-0".  Find  (1)  angle  between  side  and  end  plates,  (2)  angle  between  side  and 
bottom  plates,  (3)  angle  between  end  and  bottom  plates,  (4)  angle  which  edge 
between  side  and  end  makes  with  bottom  plate,  (5)  dimensioned  pattern  for  all 
plates,  allowing  laps  of  3"  for  riveting  as  shown. 

524.  [1]  Scale,  ^"  =  I'-O".  The  steel  plate  body  of  a  Gondola  car  is  shown 
in  Fig.  23.  Show  the  3  views  of  the  car  and  find  the  plate  dimensions,  lengths 
and  angles,  as  follows : 

Plate  A= X Plate  B= x 

Plate  C= X and  Plate  D  = x  .  . . ; . .  and 

Door  Plate  = x Angle  between  A  and  C  = ° '. 

Angle  bet.  B  and  D  = ° '.    Angle  bet.  C  and  D  = ° '. 

Length  of  intersection  between  C  and  D  = 

525.  [1]  In  the  sheet  metal  reducer  shown  in  Fig.  24,  find,  the  following: 
(a)angle  between  planes  A  and  B,  and  between  planes  B  and  C,  (b)  angles  be- 
tween planes  A  and  G,  and  B  and  F,  (c)  pattern  for  transition  section  of  the 
reducer,  dimensioning  same  in  full. 

526.  [1]  Scale,  ]|"  =  I'-O".  The  plan  and  elevation  of  a  metal  ash  shute  are 
given  in  Fig.  25,  the  dimensions  locating  all  lines  except  MN  and  NO  which  are 
to  be  found.  Lay  out  and  dimension  the  top  plate  A,  bottom  plate  C,  and  one 
side  plate  B  of  the  transition  connection.  Also  find  the  following  angles: 
(1)  Angle  between  A  and  B,  (2)  angle  between  F  and  E,  (3)  angle  between  B 
and  G. 

527.  [1]  Scale,  V  =  I'-O".  A  stone  flat  arch  over  a  window  is  shown  in  Fig. 
26.  Find  the  angles  between  the  various  faces  of  the  keystone  K,  and  develop 
this  stone,  showing  dimensions  in  full. 

528.  [1]  Scale,  V  =  I'-O".  In  the  flat  window  arch  of  Fig.  26,  find  the  angle 
between  the  various  faces  of  stone  E  and  develop  this  stone,  showing  dimensions 
in  full. 


GENERAI,   PROEI.EMS   BASED   ON   POINT,   LINE   AND  PLANE  3I 

Note.— Figure  29  shows  in  plan  and  elevation,  a  hip  and  valley  roof,  whose  solution 
may  be  taken  as  typical  of  the  work  necessary  in  the  design  of  structural  steel  roofs.  The 
following  are  the  important  angles:  (i)  the  angles  which  planes  as  R,  S,  X,  etc.,  make 
with  a  horizontal  plane,  (2)  the  angles  which  the  lines  2-r,  lo-ii,  etc.,  make  with  vertical 
hip-web  planes  through  the  hips  1-5,  11-14,  etc.  respectively,  (3)  the  angles  which  the  hip 
or  valley  rafters,  as  ic-14,  2-5,  and  7-8  make  with  a  horizontal  plane,  (4)  the  angles  in  a 
roof  plane  which  the  main  rafters  5-6  and  8-14,  etc.  make  with  the  hip  or  valley  rafters 
1*5.  7-8,  etc.,  respectively,  (5)  the  angles  between  a  vertical  line  and  the  trace  of  a  purlin 
web  upon  the  hip  web  plane  (the  purlins  have  their  web  planes  perpendicular  to  the  roof 
plane,  as  indicated  in  figure — the  line  AB  then  would  be  the  trace  of  a  purlin-web  plane 
on  the  roof  plane  R — the  hip-web  plane  is  a  vertical  plane  through  the  hip),  (6)  the  angle 
between  the  roof  plane  and  the  back  of  hip,  (7)  angle  in  the  purlin-web  between  a  normal 
to  the  center  line  of  the  purlin  and  the  trace  of  the  hip  web  on  the  purlin-web,  (8)  angle 
between  purlin-web  and  hip-web,  (9)  angle  in  back  of  hip  between  a  line  normal  to  the  hip 
web  and  the  trace  of  the  purlin-web  on  back  of  hip,  (10)  angle  between  traces  of  hip-web 
and  back  of  hip,  (11)  angle  between  back  of  hip  and  purlin-web. 

529.  [2]   Scale,  y  =  I'-O''.    Find  the  above  angles  for  the  plane  R  in  Fig.  29. 

530.  [2]    Scale,  Y'  =  I'-O".     Find  the  above  angles  for  the  plane  S  in  Fig.  29. 

531.  [2]    Scale,  Y'  =  I'-O''.     Find  the  above  angles  for  the  plane  T  in  Fig.  29. 

532.  [2]    Scale,  y  =  I'-O".     Find  the  above  angles  for  the  plane  X  in  Fig.  29. 

533.  [1]  A  carpenter's  saw-horse  is  constructed  as  shown  in  Fig.  36.  Find 
the  correct  projections  of  two  legs  as  shown,  then  detail  one  of  the  legs,  showing 
all  angles  which  would  be  used  in  cutting.     Use  scale,  3"  =  I'-O''. 

534.  [1]  Fig.  37  shows  the  center  line  of  a  4"  x  4"  brace  to  be  run  between 
two  5"  X  5''  beams,  so  that  its  top  and  bottom  faces  are  perpendicular  to  V. 
Find  the  projections  of  this  brace,  showing  its  intersection  lines  with  each  of  the 
beams  and  then  detailing  said  brace,  showing  all  angles  which  would  be  used  in 
cutting  it.    Use  scale,  |"  =  1". 

535.  [1]  Fig.  37  shows  the  center  line  of  a  4"  x  4''  brace  to  be  run  between 
two  5"  X  5"  beams,  so  that  its  two  side  faces  are  perpendicular  to  H.  Find  the 
projections  of  this  brace  showing  its  intersection  lines  with  each  of  the  beams,  and 
then  detailing  said  brace,  showing  all  angles  which  would  be  used  in  cutting  it. 
Use  scale,  |"  =  V. 

536.  [2]  Construct  the  projections  of  a  jack  rafter  B  as  shown  in  Fig.  38, 
showing  in  detail  its  intersections  with  the  hip  and  valley  rafters.  Then  find  the 
angles  which  would  be  used  in  marking  these  rafters  for  cutting. 

537.  [2]  Construct  the  projections  of  a  house  rafter  A  as  shown  in  Fig.  39, 
showing  in  detail  its  intersection  with  the  ridge  rafter  and  the  plate.  Then  find 
the  angles  which  would  be  used  in  marking  these  rafters  for  cutting. 

538.  ri]  Construct  in  detail  the  3  projections  of  the  auditorium  floor  joist  A 
shown  in  Fig.  42,  finding  the  angles  which  would  be  used  in  marking  out  ready 
for  cutting. 

539.  [1]  Construct  in  detail  the  3  projections  of  the  Auditoirum  floor  joist  B 
shown  in  Fig.  43,  then  detail  the  piece  showing  all  angles  which  would  be  used 
in  marking  out  ready  for. cutting. 

540.  [2]  Assuming  the  base  lines  in  H  for  the  face  planes  of  a  bridge  pier  as 
indicated  in  Fig.  44  and  the  respective  batirs  for  said  faces,  find  the  projections 
of  the  pier. 

541.  [2]  The  line  B(- 6'',  +  2'',  0'^  A(-2Y',  +  ¥',  + H'')  and  a  point 
M(- 3'',  +  3'', -1- 2'')  determine  the  plane  of  the  back  of  a  channel  iron  used  on  a 
hoist  boom,  the  channel  dimensions  to  the  scale  of  the  drawing  being  W  =  ly , 
F=  f,  as  in  Fig.  41.  Find  (1)  a  point  C  in  the  opposite  edge  of  the  channel 
back,  (2)  the  projections  of  the  intersection  of  the  channel  iron  with  a  horizontal 
base  plate,  (3)  the  angle  for  a  bent  plate  connection  between  the  channel  back 
and  the  base  plate  as  shown.  Then  detail  the  bent  plate,  allowing  y  clearance 
all  around. 


32  gf:ne;rai,  problems  based  on  point,  line  and  plane 

543.  [2]  The  line  B(- 3'',  +  3^",  0")  A(- 5",  +  2f",  +  3f' ')  and  a  point 
M(-2f ,  f  1'', +  3'')  determine  the  plane  of  the  web  of  an  inclined  I-beam  con- 
nection in  a  steel  structure,  the  beam  dimensions  to  the  scale  of  the  drawing  be- 
ing W-  1^-",  F  =  I'',  as  in  Fig.  40.  Find  (1)  a  point  C  in  the  opposite  edge  of 
the  back  of  the  I-beam,  (2)  the  projections  of  the  I-beam,  (3)  its  intersection 
with  a  horizontal  base  plate.  Then  detail  a  bent  plate  connection  for  the  same, 
allowing  ^"  clearance  all  round,  and  find  the  angle  at  which  it  is  to  be  bent. 

543.  [1]  A  square  butt-jointed  hopper  has  dimensions  as  shown  in  Fig.  71. 
Find  all  the  bevel  angles  necessary  for  its  construction,  and  in  a  separate  figure 
show  detailed  views  of  one  side  of  the  hopper,  giving  angles  and  lengths, 

544.  [1]  Fig.  95  shows  a  tin  furnace  pipe  transition  piece.  Find  the  angles 
which  its  various  sides  make  with  each  other  and  with  the  sides  of  the  vertical 
pipes  connected.     Then  develop  the  transition  piece,  dimensioning  in  full. 

545.  [2]  Draw  the  projections  of  the  cottage  of  Fig.  19,  and,  assuming  that 
rays  of  light  are  parallel  to  the  line  R(- 2'',  +  1^",  +  1^")  L(-  l",  +  F',  +  i"),  find 
the  shadow  which  the  cottage  casts  on  the  ground,  also  the  shadow  which  the 
gable  casts  on  the  roof.    Use  scale,  1''  =40'. 

546.  [4]  A  1|''  cube  stands  in  the  1st  quadrant  on  H,  with  t'gie  edges  of  its 
base  oblique  to  the  ground  line.  If  line  R(-  2",  +  1^",  +  1^")  L(-  1",  +  V\  +  i") 
gives  the  direction  of  rays  of  light,  find  its  shadow  upon  H. 

547.  [2]  A  regular  pyramid  (base  V  square)  is  located  in  the  3rd  quadrant, 
with  its  vertex  V'  below  H  and  its  base  in  a  plane  parallel  to  H  and  3^"  below  H. 
If  the  line  R(- 2", -:- 1|",  +  1^'')  L(- 1", +  1",  +  *")  gives  the  direction  of  rays 
of  light,  find  the  shadow  of  the  pyramid  on  the  plane  of  its  base. 

548.  [2]  Find  the  shadow  cast  by  the  pyramid  of  Problem  303  upon  a  hori- 
zontal plane  V  below  H.  The  line  R(- 2",  +  1^",  +  H")  L(- l",  +  r>  +  ¥') 
gives  the  direction  of  light  rays. 

549.  [2]  Find  the  shadow  cast  by  the  pyramid  of  Prob.  730  upon  the  plane  T 
there  given,  also  the  shadow  upon  H.     Direction  of  light  rays  is  given  bv  line 

R(-2",  +  u",  +  ir)  h(-r',  +  V',  +  ¥')- 

550.  [1]  The  point  0(- 10",  +  4^",  +  3^")  is  the  vertex  of  an  oblique  cone, 
and  the  point  M(-  13^'', -!- 2",  0")  the  center  of  its  circular  base  of  2^"  diameter 
in  H.  Find  its  shadow  upon  H  and  the  plane  T(-  9",  -  112^°,  +  30°)  if  light  ravs 
are  parallel  to  the  line  R(-  2'',  i  U",  +  H'')  L(-  1",  +  i",  f  ¥')- 

551.  [2]  An  oak  block,  cut  2|"  square  and  5''  deep,  lies  upon  H  in  the  1st 
quadrant,  with  its  vertical  faces  at  60°  and  30°  with  V  respectively.  On  top  of- 
this  block,  is  a  regular  pyramid,  altitude  2^",  base  1^"  square,  center  of  base  at 
center  of  block.  If  the 'line  R(- 2",  +  1^"',  +  U")  L(- 1", +  F'» +  i")  gives  the 
direction  of  rays  of  light,  find  the  shadow  of  these  objects  upon  H  and  upon  each 
other. 

552.  [1]  P  at  -6".  Assuming  that  rays  of  light  are  parallel  to  a  line 
R(_3''/+l.|';,  +  l|'')  L(_i'',  ^.|",  +  ^"),  find  the  shadow  of  the  Boom  Derrick 
and  guy  ropes  of  Fig.  27  upon  the  horizontal  plane. 

553.  [2]  A  2''  cube  stands  on  H  with  two  of  its  vertical  sides  making  60°  and 
30°  with  V  respectively.  On  top  of  this  cube  is  a  block  3"  square  and  V'  deep, 
with  its  center  above  that  of  the  cube  and  with  its  edges  parallel  to  the  cube 
edges.  If  light  rays  are  parallel  to  line  R(-  2",  f  H",  +  1^-")  L(-  V,  +  V',-+  V'), 
find  the  shadow  of  these  objects  upon  H  and  upon  each  other. 


33 

BUILDING  UP  SOLIDS,  CONDITIONS  GIVEN. 

Pyramids. 

559.  [1]  Find  the  H  and  V  projections  of  a  regular  triangular  (or  square  or 
hexagonal)  pyramid  whose  vertex  is  at  the  point  V(- Sf",  +  4-]-''',  +  5|")  and 
whose  base  is  in  a  plane  T  passed  through  the  point  0(-  10;^'',  +  If,  +  2|'')  par- 
allel to  K(-12",  +  4r,  +  -i")  L(-6i",  +  f",  +  2i'')  and  M(- 91", +  ^", +  43") 
N(- 6f , +  4", +  1"),  one  side. of  the  base  being  the  line  of  intersection  of  said 
plane  T  with  a  plane  passed  through  the  point  O  perpendicular  to  the  line  KL. 

5G0.  [1]  P  at  -  Ti".  Construct  the  3  projections  of  a  regular  square  pyramid 
whose  vertex  is  at  the  point  V(-  If",  -  5:]^",  -  4f")  and  whose  base,  in  a  plane  T 
passing  through  point  A(-  G",  -  23",  ~  If")  par.  to  the  lines  B(-  7f",  -  4",  -  4f") 

C(-2",-3r,-r)  and  D(- Sf",  -  41",  -  T )  E(- 21",  -  T,  -  4"),  has  as  one 
side  the  line  of  intersection  between  the  said  base  plane  and  a  plane  passing 
through  A  perpendicular  to  the  line  BC. 

561.  [1]  Find  the  H  and  V  projections  of  a  regular  triangular  pyramid  whose 
vertex  is  at  the  point  V(- 5J",  +  4f",  +  5f")  and  whose  equilateral  base  is  in  a 
plane  T  passed  through  the  point  0(- IQi",  +  If'.  +  2f")  parallel  to  the  Hue 
K(-12",  +  4f",-^4")  L(-6r,  +  r,  +  2i")  and  the  line  M(- 9f",  +  i",  +  41") 
N(- Gf",  +  4", +  1"),  one  side  of  the  base  being  the  line  of  intersection  of  said 
plane  T  with  a  plane  passed  through  the  point  O  perpendicular  to  the  H  trace  of 
the  plane  T. 

562.  [1]  P  at  -8".  A  regular  hexagonal  pyramid,  altitude  5"  has  its  base  in 
the  plane  B(-  8",  +  60°,  -  60° )  with  its  center  at  the  point  0(-  4|",  -  3",  z).  Each 
side  of  the  base  is  2"  in  length  and  its  lowest  side  makes  an  angle  of  45°  with  the 
V  trace  of  the  plane  B.    Construct  the  H,  V  and  P  projections  of  the  pyramid. 

563.  [1]  A  regular  hexagonal  pyramid  has  vertex  at  point  A(-  11|",+  5",+  5") 
and  its  base  in  the  plane  B(- 8^",  -  135°,  +  150°).  The  diameter  of  the  circle 
which  circumscribes  its  hexagonal  base  is  3"  and  two  sides  of  said  base  are  par- 
allel to  the  H  trace  of  plane  B.  Find  the  H  and  V  projections  of  the  pyramid, 
its  altitude,  and  its  pattern,  including  the  base. 

564.  [1]  P  at  -  ?4".  A  regular  pyramid  with  a  1^"  square  base  and  3^"  alti- 
tude is  located  in  the  third  quadrant.  The  square  base  is  in  the  plane 
T(-2|", +  150°,-120°),  with  two  sides  parallel  to  the  V  trace  and  with  its  cen- 
ter at  a  point  O  whose  vertical  projection  is  at  the  point  (- 5",  0",  - 1^"). 
Find  the  H,  V  and  P  projections  of  the  pyramid,  showing  visible  and  invisible 
edges. 

565.  [1]  P  at  -8".  Find  the  3  projections  of  a  regular  square  pyramid  whose 
vertex  is  at  the  point  K(- 1^",  -  If",  - 1")  and  whose  square  base  in  the  plane 
T(- 6",  +  60°,  -  45°)  is  inscribed  in  a  circle  of  3"  diameter  with  center  at 
Oi-H'',-2Y',z),  with  two  of  its  sides  parallel  to  the  H  trace  of  plane  T. 

566.  [1]  P  at  -6".  Scale,  j%-''  =  l'-0'\  The  lines  A(- 35',  - 18'.  -  5') 
D(-  18',  -  27',  -  5'),  a  line  parallel  to  AD  through  B(-  25',  -  2V,  -  20'),  and  AB 
are  respectively  the  ridge,  eave  and  front  rafter  lines  of  a  cottage  roof.  A  tower 
with  a  pyramidal  roof  is  to  be  built  thereon,  the  base  of  the  tower  on  the  roof 
to  be  an  8'-0"  square,  whose  sides  are  respectively  parallel  to  the  lines  AB  and 
BC  and  whose  center  is  at  distances  14'-3"  and  12'-0"  respectively  therefrom. 
The  top  of  the  pyramidal  roof  is  to  be  12'-6"  directly  above  this  center,  and  the 
4  vertical  walls  of  the  tower  rise  to  a  horizontal  plane  7'-6"  above  the  center  of 
the  square  in  the  roof.  The  four  corner  tower  rafters  are  to  be  9'-0"  in  length. 
Find  the  3  projections  of  the  roof  and  tower  theron,  the  distance  of  the  top  of 
the  tower  from  the  plane  of  the  roof  and  the  angle  which  the  corner  tower  rafters 
make  with  each  other. 


34  BUIIvDING  UP  SOIJDS,  CONDITIONS  GIVEiN 

Cones. 

567.  [2]  Construct  the  H  and  V  projections  of  a  right  circular  cone  whose 
base  of  IV'  diameter  is  in  the  plane  T(- 5^'',  -  60°,  +  30°)  with  center  at  the 
point  0(-  3f' ,  +  If",  z).    Altitude  of  cone  is  3". 

568.  [2]  The  point  A(- 6'',  -  4|",  -  5")  is  the  vertex  of  a  cone  of  revolution 
whose  axis  is  perpendicular  to,  and  whose  base  of  2^^  diameter  is  in,  the  plane 
T(- 6",  +  45°,  -  45°).  Find  the  H  and  V  projections  of  this  cone.  What  is  its 
altitude  ? 

569.  [1]  P  at-7|".  A  cone  of  revolution  with  a  base  of  2"'  diameter  and 
3|''  altitude  is  located  in  the  3rd  quad.  Its  base  is  in  plane  T(-  2^',  +  150°,  -  120°) 
with  its  center  at  the  point  0(-  5",  y,  -  1^'')-    Find  the  3  projections  of  this  cone. 

Prisms. 

570.  []]  P  at  -  SV\  Construct  the  3  projections  of  a  regular  hexagonal  prism 
in  the  3rd  quadrant  3^"  high,  the  plane  of  whose  base  makes  an  angle  of  30° 
with  V  and  75°  with  H.  Each  side  of  the  base  is  1"  long  and  two  edges  are  par- 
allel to  V.    The  center  of  the  base  is  4"  to  left  of  P,  f "  behind  V  and  1^"  below  H. 

571.  [1]  P  at  -  2".  Construct  the  H,  V  and  P  projections  of  a  regular  square 
prism -in  the  1st  quadrant,  altitude  3^",  each  side  of  base  If",  center  of  base  at 
point  C(-9i",  +  l|'',  +  f")-  Plane  of  base  makes  an  angle  of  30°  with  H  and 
75°  with  V,  two  edges  of  base  being  parallel  to  H. 

572.  [1]  P  at  -  8y'.  A  regular  hexagonal  prism  in  the  third  quadrant,  of  2" 
altitude,  one  hexagonal  base  in  the  plane  T(- 4^",  t- 45°,  -  60°),  with  center  at 
point  0(- 2'',  -  1^",  z")  and  each  side  ly,  is  hollowed  out  by  a  right  circular 
cylinder  of  1^"  diameter  whose  axis  coincides  with  that  of  the  prism.  Construct 
the  H,  V,  and  P  projections  of  the  hollow  prism. 

Cubes. 

573.  [1]  P  at  -3".  A  2''  cube  stands  on  the  plane  T(- 8",  -  30°,  +  60°)  in 
the  first  quadrant.  The  center  of  its  base  when  revolved  into  H  about  the  H 
trace  of  plane  T  is  at  0(-  7|",  -(-  2|'',  0")  and  two  of  the  sides  of  the  base  make 
angles  of  36°  with  this  trace.     Find  the  3  projections  of  the  cube. 

574.  [1]  G.  L.  par.  to  short  edge  of  sheet.  P  at  -54''.  A  cube  in  the  third 
quadrant  has  its  upper  face  in  the  plane  T(- 4^",  +  60°,  -  45°),  one  side  of  this 
face  being  the  line  A(-  2V',  y",  -  If')  B(-  i",  y",  -  3f")  in  the  plane  T.  Find 
the  3  projections  of  this  cube, 

575.  [1]  The  points  A(- 11", +  |",  0"),  B(- 8", +  i",  0"),  C(- 8",  +  3^",  0") 
are  three  vertices  of  the  base  of  a  cube  standing  in  the  1st  quadrant  on  H.  Re- 
volve the  cube  about  the  H  trace  of  the  plane  T(- 12^",  -  45°,  + 30°)  until  it 
stands  on  T,  showing  its  H  and  V  projections  in  this  position. 

576.  [1]  G.  L.  par.  to  short  edge  of  sheet.  A  3"  cube  stands  in  the  first  quad- 
rant with  one  of  its  diagonals  perpendicular  to  H,  and  another  parallel  to  V. 
Find  the  projections  of  the  cube,  and  the  distance  from  one  vertex  to  the  plane  of 
the  3  adjacent  vertices. 


REPRESENTATION  OF  SURFACES. 


HELICAL  CONVOLUTES. 

Assume  elements;  intersection  with  H  or  oblique  plane;  assume  points  on 
surface. 

578.  [2]  A  Helical  Convolute  is  formed  by  drawing  tangents  to  a  helix  of  3'' 
pitch  in  the  first  quadrant  whose  axis  is  perpendicular  to  H  through  the  point 
0(-  6^",  +  11",  0")  and  whose  generating  point  starts  at  pt.  M(-  7^'',  +  1^",  0") 
moving  so  its  H  projection  appears  to  rotate  counter-clockwise.  Find  16  elements 
in  the  first  convolution  of  the  surface,  the  line  in  which  the  surface  thus  determined 
cuts  H,  and  the  projections  of  a  point  K  of  the  surface  not  on  one  of  the  above 
elements. 

579.  [1]  G.  L.  par.  to  short  edges  of  sheet.  A  Helical  Convolute  has  as  its 
directrix  a  helix  whose  axis  is  M(- 7|",  +  2",  0")  N(-7i",  +  2",  +  5"),  whose 
pitch  is  12'',  whose  generating  point  starts  at  K(-  9",  +  2",  0")  and  moves  so  that 
its  H  projection  appears  to  rotate  counter-clockwise.  Find  the  intersection,  (be- 
tween H  and  the  point  where  the  helical  directrix  pierces  T),  of  the  helical  con- 
volute and  a  plane  T(- 1-^',  -  120°,  +  150°). 

HYPERBOLIC  PARABOLOIDS  AND  CONOIDS. 

Assuming  elements,  first  and  second  generations;  plane  directors  of  both 
generations;  assuming  points  on  surface. 

580.  [4]  Plane  T(-3f',- 15°, +  224°)  is  plane  directer,  A(- 7t'',  +  f",  +  2f' ) 
B(-6Y',  +  2V\0'')  and  C(-4r,  +  2f',  +  3")  D(- Sf",  +  f",  4  2'')  the  directrices 
of  a  Hyperbolic  Paraboloid.  Find  the  projections  of  an  element  of  the  surface 
parallel  to  the  line  E(-  2|",  0",  z)  F(-  1V\  +  i",  z)  lying  in  the  plane  directer. 

581.  [4]  In  the  Hyperbolic  Paraboloid  of  Prob.  580,  find  an  element  of  the 
surface  through  the  point  K(-  7|'',  y,  z)  on  the  directrix  AB. 

582.  [4]  A  conoid  has  plane  T(- 3^',  +  30°,  -  30°)  as  a  directer  and  line 
^(-^V',-i'',-2^/'-)  B(-6'', -2", -I")  and  arcs  of  2^'  radius  struck  to  the 
right  of  the  projections  of  B  as  centers  for  directrices.  Find  an  element  of  the 
conoid  parallel  to  line  C(-  3",  -  -J",  z)  D(-  f ,  -  f ,  z)  lying  in  the  plane  directer. 

583.  [4]  In  the  Conoid  of  Prob.  582,  find  an  element  of  the  surface  through 
the  point  K(-  7",  y,  z)  on  AB. 

584.  [2]  Given  the  plane  directer  T(-  3",  +  18°,  -  110J-°)  of  a  Hyperbolic  Par- 
aboloid, with  right  line  directrices  A(-  7f' ',  -  2'',  -  3")  B(-  6f' ,  -  Y',  - 1")  and 
C(-5V',-lV',-^¥')  D(-4",-2f' ,-|").  Find  the  9  elements  of  the  surface 
which  divide  the  directrix  AB  into  8  equal  parts.  Find  a  point  E  on  the  surface 
whose  H  projection  is  the  point  (- 5'',  -  2'',  0"). 

585.  [2]  The  lines  M(- 6",  +  f ',  O'O  N(- 4'',  +  2'',  +  2''),  P(- 2'',  +  1'',  0'') 
Q(- 1",  -  IJ'',  +  If")  are  the  directrices  of  a  Hyperbolic  Paraboloid  whose  plane 
directer  is  H.  Asume  10  elements  of  the  surface  and  find  a  point  E  on  the  sur- 
face whose  H  projection  is  the  point  (-4:1",  +  1|",  z"). 

586.  [2]   In  the  Hyperbolic  Paraboloid  of  Prob.  585,  find  the  plane  directer  of 


36  REPRESENTATION  OF  SURFACES 

the  second  generation,  ten  elements  of  the  second  generation,  and  a  point  A  on 
the  surface  whose  Y  projection  is  (- 3|'',  y,  +  f). 

587.  [1]  The  two  right  Hnes  M(-  13^",  +  U",  +  3^')  N(-  11^",  +  1",  0")  and 
P(-8J'',  0",  +  3")  Q(-5i",  +  5i",  0")  are  the  directrices  of  Hyperbolic  Parabo- 
loid, whose  plane  directer  is  D(-  I31",  -  150°,  +  120°).  Find  (1)  five  elements  of 
the  first  generation  through  points  M,  A,  B,  C,  N  which  divide  the  directrix  MN 
into  4  equal  parts,  (2)  plane  directer  of  the  second  generation,  (3)  five  elements 
of  the  second  generation. 

588.  [1]  The  two  right  lines  M(- 13i",  +  4i",  +  3i")  N(- 11^",  +  1",  0")  and 
P(-8|''",  0",  f  3")  Q(-5|",  +  5y',  0'')  are  the  directrices  of  a  Hyperbolic  Para- 
boloid, whose  plane  directer  is  D(-  K3^'',  -  150°,  +  120°).  Find  (1)  nine  elements 
of  the  first  generation  through  points  on  the  directrix  MN  which  divide  it  into  8 
equal  parts,  (2)  the  vertical  projection  of  a  point  O  in  the  surface  whose  H  pro- 
jection is  at  the  point  (- 11",  +  2",  0"),  (3)  an  element  of  the  2nd  generation 
through  this  point  O. 

589.  [1]  The  two  right  lines  M(- 12",  -  5f',  -  |")  N(- 6^",  -  3",  -  4^")  and 
P(-llJ:",-l",-4")  Q(-Tf' ,-l",-ir)  are  the  directrices  of  a  Hyperbolic 
Paraboloid.     The  lines  MQ  and  PN  are  elements  of  the  first  generation.     Find 

(1)  eight  elements  of  the  first  generation  through  points  P,  A,  B,  C,  D,  E,  F,  Q 
which  divide  the  directrix  PQ  into  7  equal  parts,  (2)  plane  directers  D^  and  Dj 
of  the  first  and  second  generations,  (3)  an  element  of  the  second  generation 
through  a  point  K(- 8",  y",  -  3^")  on  the  surface. 

590.  [1]  The  two  right  lines  M(- 12^",  -  5",  -  3")  N(- 12^",  0",  -  3")  and 
P(-  8|",  0",  -  IV')  Q(-  5i",  -  5",  -  4^")  are  the  directrices  of  a  Hyperbolic  Par- 
aboloid whose  plane  directer  is  V.  Find  (1)  five  elements  of  the  first  generation 
through  points  M,  A,  B,  C,  N  which  divide  the  directrix  MN  into  4  equal  parts, 

(2)  plane  directer  for  the  second  generation,  (3)  an  element  of  the  second  gen- 
eration through   a   point   on   the   surface   whose   H   projection   is   at   the   point 

(-9r,o",-2r). 

591.  [1]  Lines  M(- 12",  0",-l")  N(- 9",  -  3^",  -  3^")  and  P(- 7",  0",  -  2") 
Q(-  5^",  -  3i",  -  2Y')  are  the  directrices  of  a  Hyberbolic  Paraboloid  whose  plane 
directer  is  V.  Find  (1)  the  9  elements  of  the  first  generation  which  divide  the 
directrix  MN  into  8  equal  parts,  (2)  the  plane  directer  of  the  second  generation, 

(3)  the  H  projection  of  the  point  K(- 9^",  y,  -  1|")  on  the  surface,  (4)  an  ele- 
ment of  the  2nd  generation  through  this  point  K. 

592.  [2]  Lines  A(- 6^",  -  J",  -  i")  B(-5",-3V4")  and  C(-3ViV3i") 
D(-y, -4",  -  2")  are  directrices  of  a  Hyperbolic  Paraboloid  of  which  BC  and 
AD  are  elements  of  the  first  generation.  Find  13  elements  of  each  generation, 
and  a  plane  directer  for  each  generation,  through  convenient  points. 

593.  [2]  Lines  A(- 6|",  -  y',  -  i")  D(- i",  -  4",  -  2"),  B(- 5",  -  2",  -  4") 
Cf- 3",  -  i",  -  3|")  are  directrices  of  a  Hyperbolic  Paraboloid  of  which  AB  and 
CD  are  elements.  Find  seventeen  other  elements  of  the  same  generation,  plane 
directers  for  the  first  and  second  generation  through  convenient  points,  the  ver- 
tical projections  of  a  point  M(- 3|",  -  If",  z)  on  the  surface,  and  an  element  of 
the  second  generation  through  this  point  M. 

594.  [2]  Lines  A(- 6|",  -  ^",  -  D  B(- 5",- 2",- 4^')  and  C(- 3".- i",- 3^") 
D(-^'',-4",  -  2")  are  directrices  of  a  Hyperbolic  Paraboloid  of  which  AC  and 
BD  are  elements.  Find  thirteen  other  elements  of  this  same  generation,  a  plane 
directer  for  the  second  generation  through  the  point  K(-  1|",  -  i",  -  ^"),  and  the 
point  on  the  surface  whose  V  projection  is  at  (-  3j[",  0",  -  2f"). 


HYPERBOLOIDS  OF  ONE  NAPPE,  ETC. 

Assuming  elements  through  given  points. 

595.  [2]  A  Hyperboloicl  of  one  Nappe  has  as  directrices  the  three  given 
right    Hnes    A(- 6",  -  |",  -  2^'')     B(- 5'',  -  2'',  -  ^'0    and    C(- 4^",  -  3",  -  2^") 

D(-3|",-i",-r)  and  E(-3^-ir,--r)  F(- 1",  -  3",  -  3")-"  Find  an  ele- 
ment of  the  surface  through  the  point  G(- 5i^",  y",  z")  on  AB ;  also  an  element 
through  the  point  F. 

596.  [2]  A  Hvperboloid  of  one  Nappe  has  as  directrices  the  3  right  lines 
K{-n",+  \"&n"^  B(-.3i'',+  2Vi"),  C(-4",  +  3",  +  2i")  D(-5",  +  i",  +  i") 
and  E(-  5-^",  +  1]",  -i-  \")  F(-  7|-'',  +  3",  +  3'').  Find  an  element  of  the  surface 
through  M(-  IJ'',  y,  z)  on  CD;  also  an  element  through  the  point  F. 

597.  [4]  The  directrices  of  a  Hyperboloid  of  one  Nappe  are  the  right  lines 
A(-7^i2Vi'0  B(-6",+  iVir)  C(-5r,+  HVlD  D(-5",  +  r,  +  |") 
and  E(-4f' ,  0",  +  2f'')  F(- 4", -;- 2", +  ^'0-  Find  an  element  of  the  surface 
through  the  point  M(-  ^\" ,  y,z)  on  AB. 

598.  [4]  A  warped  surface  has  as  directrices  the  right  line  A(-  4f",-  2|",-  ^") 
B(-5^'',  -  f', -2|"),  arcs  of  ?.\"  radius  struck  to  the  left  with  the  projections 
of  B  as  centtrs,  and  arcs  of  2|"  radius  struck  to  the  right  with  the  projections  of 
A  as  centers.  Find  an  element  of  the  surface  through  the  point  C(-  5^",  y" ,  z") 
in  line  AB. 

599.  [4]  A  warped  surface  has  as  directrices  right  lines  K{- \\" ,\'iV\^%"^ 
B(-4r',  +  li",  +  ir)  and  C(- 6^',  +  2f',  +  l-J")  D(-7r,^-r.  +  D  and  arcs 
of  2|"  radius  struck  to  the  right  with  the  projections  of  A  as  centers.  Find  an 
element  through  point  B. 

HYPERBOLOIDS  OF  REVOLUTION  OF  ONE  NAPPE. 

Assume  elements;  second  generations;  construction  of  meridian  curves; 
assume  points  on  surface. 

GOO.  [1]  The  line  A(- 9",  -  3^',  -  2]")  B(- 9",  0",  -  2^")  is  the  axis  of  a 
Hyperboloid  of  Revolution  of  one  Nappe,  located  in  the  third  quadrant  with  its 
base  in  Y.  The  generatrix  in  its  initial  position  is  M(- 11",  0",  -  If") 
N(- 9",  -  If",  -  If").  Find  (1)  thirty-two  elements  of  one  generation  of  the 
surface,  which  pierce  V  at  equal  intervals  around  the  circumference  of  its  base, 
(2)  the  horizontal  projection  of  the  medirian  section  parallel  to  H,  (3)  an  ele- 
ment of  each  generation  through  a  point  K  whose  V  projection  is  not  in  the  V 
projection  of  any  of  the  above- 32  elements. 

601.  [1]  The  line  A(- 11",  +  2f",  +  5")  B(- 11",  +  2f",  0")  is  the  axis  of  a 
Hyperboloid  of  Revolution  of  one  Nappe  which  stands  in  the  first  quadrant  with 
its  base  in  H.  The  generatrix  in  its  initial  position  is  M(- 134",  +  2",  0") 
N(- 11",  +  2",  +  2i").  Find  (1)  sixteen  elements  of  each  generation  of  the  sur- 
face, which  pierce  H  at  equal  intervals  around  the  circumference  of  its  base, 
(2)  the  vertical  projection  of  the  meridian  section  parallel  to  V,  (3)  the  vertical 
projection  of  a  meridian  section  making  45°  with  V. 

602.  [2]  The  line  A(-  5",  -  2",  -  4")  B(-  5",  -  2",  0")  is  the  axis  of  a  Hyper- 
boloid of  Revolution  of  one  Nappe,  located  in  the  third  quadrant  with  its  upper 
and  lower  bases  respectively  in  H  and  in  a  horizontal  plane  4"  below  H.  The  gen- 
eratrix of  this  surface  is,  in  its  initial  position,  the  line  M(- 6|",  -  1|",  -  4") 
N(-  5",  -  \V' ,  -  2").  Find  (1)  thirty-two  elements  of  one  generation  of  the  sur- 
face which  pierce  H  at  equal  intervals  around  the  circumference  of  its  base, 


38  REPRESENTATION  OE  SURFACES 

(2)  the  vertical  projection  of  the  meridian  section  parallel  to  V,  (3)  the  vertical 
projection  of  a  meridian  section  making  45°  with  V. 

603.  [3]  A  Hyperboloid  of  Revolution  of  one  Nappe  stands  in  the  first  quad- 
rant with  its  base  of  4'"  diameter  in  H  with  center  at  the  point  0(-  4|'',  +  2^'',  0"). 
The  circle  of  the  gorge,  of  2V'  diameter,  is  on  a  horizontal  plane  1^'  above  H, 
with  its  center  directly  above  O.  Find  (1)  thirty-two  elements  of  the  surface, 
(2)  the  vertical  projection  of  the  meridian  section  parallel  to  V,  (3)  an  element 
of  each  generation  through  a  point  K  whose  H  projection  is  not  on  the  H  projec- 
tion of  any  of  the  thirty-two  elements  assumed  above. 

604.  [2]  A  portion  of  a  Hvperboloid  of  Revolution  of  one  Nappe  is  generated 
by  the  revolution  of  the  line'C(- 4^'',  +  4",  +  2|'0  D(- 2",  +  2^",  0'')  about  the 
axis  A(-  4",  +  2V',  0")  B(-  4'',  +  2|",  +  4|")-  Show  32  elements  of  this  surface 
and  determine  accurately  the  projections  of  the  meridian  curve  cut  from  the  sur- 
face by  a  meridian  plane  parallel  to  V. 

605.  [1]  G.  L.  par.  to  short  edge  of  sheet.  A  portion  of  a  Hyperboloid  of  Rev- 
olution of  one  Nappe  is  generated  by  the  revolution  of  line  M(- 7|",  +  If",  +  f ) 
N(-5J-", +  4^'',-!  5D  about  the  line  A(- 5i'',  +  3",  +  f  0  B(- 54",  +  3",  +  5^:") 
as  an  axis.  Find  16  elements  of  each  generation  of  the  surface  and  determine 
accurately  the  projections  of  the  meridian  section  parallel  to  V. 

HELICOIDS. 

Oblique  and  Right;  assume  elements;  intersections  with  H  or  V;  assume 
points  on  surface. 

606.  [2]  The  line  A(- 4",  +  2",  +  4")  B(- 4",  +  2",  0")  is  the  axis  of  an 
oblique  Helicoid ;  the  generatrix  in  initial  position  is  M(- 5^",  +  2",  0") 
N(-4", +  2",  +  1|")  which  moves  in  such  a  way  that  its  horizontal  projection  ap- 
pears to  rotate  counter-clockwise.  The  pitch  of  the  helix  which  is  generated  by 
the  point  M  is  2".  Find  (1)  sixteen  elements  of  the  surface  generated  during  one 
complete  movement  of  the  generatrix  about  the  axis,  (2)  the  intersection  of  the 
surface  thus  determined  with  the  H  plane  of  projection,  (3)  the  two  projections 
of  a  point  K  which  is  not  on  any  of  the  elements  assumed  above. 

607.  [2]  The  line  A(- 4^,  -  If",  -  4")  B(- 4^,  -  IF,  0")  is  the  axis  of  an 
oblique  Helicoid,  whose  generatrix  in  initial  position  is  M(- 5^",  -  If",  -  4") 
N(-4y,  -  If",  -  If").  This  generatrix  moves  in  such  a  way  that  its  H  projec- 
tion appears  to  rotate  clockwise  and  the  point  M  generates  a  helix  whose  pitch 
is  2|".  Find  (1)  sixteen  elements  of  the  surface  generated  during  one  complete 
movement  of  the  generatrix  about  the  axis,  (2)  the  intersection  of  the  surface 
thus  determined  with  a  horizontal  plane  4"  below  H,  (3)  the  projections  of  a 
point  K  of  the  surface  not  on  one  of  the  above  elements. 

608.  [2]  The  line  A(- 5",  -  4",  -  2")  B(- 5",  0",  -  2")  is  the  axis  of  a  Right 
Helicoid  whose  generatrix  in  its  initial  position  is  determined  by  the  points 
M(-  3|",  0",  -  2")  and  N(-  5",  0",  -  2").  This  generatrix  moves  in  such  a  way 
that  its  V  projection  appears  to  rotate  clockwise,  and  the  point  M  generates  a 
helix  whose  pitch  is  2".  Find  (1)  thirty-two  elements  of  the  surface,  generated 
while  the  generatrix  swings  about  the  axis  twice,  (2)  the  projections  of  another 
helicoid,  exactly  similar  to  the  first  and  4"  above  it. 

609.  [2]  The  line  A(-  4",  -  2V',  0")"B(-  4",  -  2^",  -  5")  is  the  axis  of  a  Right 
Helicoid,  whose  generatrix  in  its  initial  position,  is  determined  by  the  points 
M(-6|",-24",-5")  and  N(- 4",  -  2^',  -  5").  This  generatrix  moves  in  such 
a  way  that  its  H  projection  appears  to  rotate  counter-clockwise,  while  the  point 
M  generates  a  helix  whose  pitch  is  2^".  Find  (1)  sixty- four  elements  of  the 
surface,  generated  while  MN  swings  about  the  axis  twice,  (2)  the  projections  of 


OP   THE 

I^NIVERSITY 

OF 

TSENTATION  OF  SURFACES  39 

another  right  hehcoid,  with  same  axis,  same  pitch,  generatrix  starting  from  same 
initial  position,  but  being  only  1^"  in  length. 

610.  [1]  G.  L.  par.  to  short  edges  of  sheet.  An  oblique  Helicoid  is  in  the  first 
quadrant,  with  a  vertical  axis  through  the  point  0(- 5^',  +  2",  0").  Its  genera- 
trix moves  from  its  initial  position  A(-  7",  -  2",  0")  B(-  5^",  -  3",  +  2V')  so  that 
its  H  projection  appears  to  rotate  counter-clockwise,  while  the  point  A  generates 
a  helix  of  8"  pitch.  Find  (a)  12  elements  of  the  surface  generated  in  f  of  a  rev- 
olution, (b)  the  intersection  of  the  surface  thus  determined  with  H,  (c)  the  pro- 
jections of  the  helices  generated  by  the  two  points  which  divide  the  generatrix 
into  3  equal  parts. 

611.  [2]  The  line  M(  -  3f ',  -  If ,  0")  N(-3r',  +  ir.^  ^")  is  the  axis  of  an 
oblique  Helicoid  whose  elements  make  an  angle  of  45°  with  H.  In  initial  position 
the  generatrix  pierces  H  at  K(- 2^",  +  If ',  0")  and  K  appears  to  rotate  clock- 
wise. Find  (1)  1()  elements  of  the  surface  formed  in  one  revolution,  (2)  the  in- 
tersection of  the  surface  thus  determined  with  the  H  plane. 

612.  [2]  An  oblique  Helicoid  has  as  axis  the  line  M(- 4|",  0",  -  2")  and 
N(- -ll'',  -  4^",  -  2"),  its  elements  making  30°  with  this  axis.  In  initial  position 
the  generatrix  pierces  H  at  the  point  K(-  6",  0",  -  2)  and  in  moving  K  generates 
a  helix  of  4"  pitch.  Find  16  elements  of  the  first  convolution,  and  the  curve  in 
which  the  surface  thus  determined  cuts  V. 

613.  [4]  An  oblique  helicoid  has  as  axis  vertical  line  through  N(-  4",  +  1",  0''), 
its  elements  making  22^°  with  the  H  plane.  In  initial  position  the  generatrix 
pierces  H  at  the  point  K(- 3",  f  2",  0")  and  in  moving  K  generates  a  helix  of 
2"  pitch.     Find  16  elements  of  the  first  convolution  of  the  surface. 

APPLICATIONS. 

614.  Fig.  50  shows  a  square  threaded  screw,  formed  by  the  motion  of  a  square 
about  a  line  in  its  plane  as  axis.  Each  point  in  the  square  generates  a  helix.  The 
top  and  bottom  thread  surfaces  are  right  helicoids,  limited  by  the  blank  and  root 
cylinders.  Find  the  projections  of  such  threads,  showing  clearly  the  method  of 
construction,  with  the  following  dimensions : — 

1.  [8]  D  =  ir-  p  =  y'. 

2.  [8]   D  =  2  ".     P  =  f". 

3.  [4]   G.  h.  par.  to  short  edge  of  space.     D  =  3  ".     P  =  1^". 

4.  [4]   G.  L.  par.  to  short  edge  of  space.     D  =  2Y'.     P  =    f . 

615.  In  Fig.  51  is  shown  a  "V"  thread.  The  two  surfaces  are  oblique  heli- 
coids and  the  threads  may  be  thought  of  as  having  been  generated  by  the  motion 
of  a  triangle  about  a  right  line  in  its  plang  as  axis,  each  point  generating  a  helix. 
Such  threads  or  modifications  thereof  are  common  on  bolts,  screws,  etc.  Find 
the  projections  of  such  threads,  showing  clearly  the  method  of  construction,  with 
the  following  dimensions  : — 

Thread  angle  =  60°. 
Thread  angle  =  90°. 
Thread  angle  =  60°. 
Thread  angle  =  90°. 

616.  In  Fig.  52  is  shown  a  round  helical  spring  generated  by  moving  a  circle 
about  a  line  in  its  plane  in  such  a  way  that  all  its  points  generate  helices  of  the 
same  pitch.  Find  the  projections  of  such  a  spring,  using  the  following  dimen- 
sions : — 

1.  [8]   D  =  1V'.     d  =  r.     P=l  ". 

2.  [8]    D  =  2"".     d  =  y'.     P  =  H". 

3.  [2]   D  =  4  ''.     d  =  l".     P  =  2  ". 


1. 

[8] 

D  =  ir. 

P  =  1" 

2. 

[8] 

D  =  2  ". 

p  =  f : 

3. 

[2] 

D  =  2y\ 

p  =  r 

4. 

[2] 

D  =  3  ''. 

P  =  2", 

40  ri;pre;sentation  of  surfaces 

G17.  In  Fig.  53  is  shown  a  square  helical  s])ring,  formed  by  the  motion  of  a 
square  about  a  line  in  its  plane  as  an  axis,  each  point  generating  a  helix  of  the 
same  pitch.  Find  the  projections  of  such  a  spring,  using  the  following  dimen- 
sions : — 

1.  [8]  D  =  i-Y'.  p = f ''.  s  =  -r. 

2.  [8]   D  =  2  ''.     P  =  1  ''.     S  =  V'. 

3.  [2]   D  =  3  ''.     P  =  1J".     S  =  r. 

618.  In  Fig.  54  is  shown  a  helicoidal  conveyor,  whose  flights  are  right  heli- 
coidal-  surfaces.  Neglecting  the  thickness  of  the  flight,  find  the  projections  of 
such  a  conveyor,  using  the  following  dimensions,  and  scale,  1"  =  I'-O". 

1.  [8]'d  =  15''.     d  =  3'^     P  =  9''. 

2.  [8]   D  =  2'-0^     d  =  6".     P  =  12". 

3.  [2]   D  =  2'-(r'.     d  =  12''.     P  =  2'-0''. 

619.  In  Fig.  56  is  shown  a  wood  screw  thread  formed  by  2  helical  convolute 
surfaces,  the  large  spaces  being  left  between  threads  to  make  up  for  the  small 
shearing  strength  of  wood  as  compared  to  the  metal  used.  Find  the  projections 
of  such  a  screw  thread,  using  the  following  dimensions : — 

1.  [8]   D  =  2''.     d  =  li''.     P  =  H''. 

2.  [8]   D  =  r\     d  =  lV'.     P  =  l'''. 

3.  [2]  D  =  3^     d  =  2X     P  =  2  ''. 

620.  [1]  Fig.  57  shows  a  right  helicoidal  stairway  built  in  a  square  tower,  with 
a  cylindrical  sheet  metal  well.  Assume  reasonable  dimensions  and  a  convenient 
scale,  and  construct  the  detailed  projections  of  such  a  stairway,  showing  all  nec- 
essary construction  lines. 

621.  [1]  Fig.  58  shows  a  right  helicoidal  stairway  designed  for  a  cylindrical 
corner  tower  in  a  gymnasium,  with  a  cylindrical  well.  Asume  reasonable  dimen- 
sions and  a  convenient  scale,  and  construct  the  detailed  projections  of  such  a 
stairway,  showing  all  necessary  construction  lines. 


PLANES  TANGENT  TO  SURFACES. 

Through  points  on  surface ;  through  points  off  the  surface ;  parallel  to  given 
lines;  through  given  lines. 

1.— PLANES  TANGENT  TO  RIGHT  CYLINDERS. 

625.  ["3]  Pass  a  plane  tangent  to  the  right  cylinder  whose  base  is  a  ?>"  circle 
with  center  at  0(- 4-^",  +  2^',  0")  and  lying  in  the  H  plane. 

(a)  Through  the  point  K(- 3^",  y,  +  2^'')  on  the  surface. 

(b)  Through  the  point  M (-  hV ,  +  f ",  +  3'') ,  not  on  the  surface. 

( c )  Parallel  to  the  line  Q  ( -  6'',  +  2|'',  +  f ' )  R  ( -  2|'',  +  4f' ,  +  4'' )  • 

620.  [2]  Pass  a  plane  tangent  to  the  right  cylinder  whose  base  is  in  the  V 
plane  and  is  the  ellipse  whose  major  axis  is  3"  long,  makes  -45°  with  the  ground 
line  and  is  intersected  at  point  B(- 4:|",  0",  -  2|")  by  the  minor  axis,  which  is 
2''  long. 

(a)  Through  a  convenient  point  M  on  the  surface. 

(b)  Through  an  assumed  point  N  not  on  the  surface. 

(c)  Parallel  to  the  line  G(- 6f ',  -  31",  -  3'0    H(- 3^'',  -  1",  -  l^'')- 

627.  [4]  P  at  -  4".  Pass  a  plane  tangent  to  the  right  cylinder  whose  base  is 
a  ?^\"  circle  lying  in  P  with  center  at  K(0",  -  \\" ,  -  \\"). 

(a)  Through  point  \^{- \\" ,  -  \" ,  z)  on  the  surface. 

(b)  Through  point  N(- 3'',  + f"'',-^''),  off  the  surface. 

(c)  Parallel  to  line  0(- 2",  -  f',  -  ^'0    Q(-|",  -  H",  -  1|"). 

628.  [2]  Pass  a  plane  tangent  to  the  right  cylinder  whose  base  is  a  2^'  circle 
lying  in  the  H  plane  with  center  at  M(-  5:1:",  -  3^'',  0"). 

(a)  Through  a  convenient  point  A  on  the  surface. 

(b)  Through  point  R(-  2f",  -  V\  -  \\"),  off  the  surface. 

(c)  Parallel  to  line  S(-  6^'^  -  4",  -  3i")   T(-  3-]-",  -  l^'',  -  I"). 

2.— PLANES  TANGENT  TO  OBLIQUE  CYLINDERS. 

629.  [2]  Pass  a  plane  tangent  to  the  cylinder  having  line  A(-  3|'',  +  3",  +  2^") 
B(- 5'',  +  2^'',  0")  for  its  axis  and  its  base  in  H,  a  circle  of  ?>"  diameter  with 
center  at  ?j. 

(a)  Tbrough  the  point  0(- 3'',  !- 2f' ',  z)  on  the  surface  of  the  cylinder. 

(b)  Through  the  point  C(-  1^",  +  3'',  +  If"),  not  on  the  surface.' 

(c)  Parallel  to  the  line  Q(-l-K',  +  2f',  0'')  R(-2'^  +  ll'^  +  3'')• 
630.      [2]    Pass  a  plane  tangent  to  the  oblique  cylinder  whose  axis  is  the  line 

K(-5i",  0'',-U-")  L(-2i",-2r,-3'')  and  whose  base  in  V  is  an  ellipse 
whose  major  axis  is  4''  long,  makes  an  angle  of  -  30°  with  ground  line  and  is  in- 
tersected at  pt.  K  by  the  minor  axis  which  is  2"  long. 

(a)  Through  a  point  A  on  the  surface. 

(b)  Through  point  M(- If",  -  If",  -  If '),  not  on  the  surface. 

(c)  Parallel  to  line  N  (- 6f",  -  2|",  -  |")    R(- 4V',  -  1^",  -  3"). 

631.  [2]  P  at  -  3^'.  Pass  a  plane  tangent  to  the  cylinder  whose  base  is  a  2" 
circle  lying  in  P  with  its  center  at  A(0",  -  If ,  - 1^")  and  having 
AB(-4i",-2t", -23")   for  its  axis: 

(a)  Through  point  C(- 2^1",  -  1-J",  z)   on  the  surface. 

(b)  Through  point  D(-  1^",  -  3",  -  1^"). 

(c)  Parallel  to  line  E(-  If",  -  2^",  +  f")  F(-  3|",  -  \l" ,  -  2f' )• 


42 

632.  [2]  Pass  a  plane  tangent  to  the  cylinder  whose  base  is  a  2V  circle  lying 
in  H  with  its  center  at  G(-  H'',-  2^,  0"),  whose  axis  is  CxH(-  If'',-  4^,-  3f"). 

(a)  Through  point  K(-  3i",  y,  -  If")  on  the  surface. 

(b)  Through  point  L(-  U",  -  2'',  -  2f"). 

(c)  Parallel  to  line  M(- 6^, +  1^  -  1")   N(- 3r,  -  3r,  -  iD- 

3.— PLANES  TANGENT  TO  RIGHT  CONES. 

633.  [4]  Pass  a  plane  tangent  to  a  right  cone  whose  base  is  a  If  circle  lying 
in  H  with  center  at  0(- 5a",  +  If",  0")  and  whose  altitude  is  2^". 

(a)  Through  point  M(- 5^',  y,  +  f")   on  the  surface. 

(b)  Through  point  N  (-  4^",  +  f ",  +  |"),  not  on  the  surface. 

(c)  Parallel  to  line  K(- 31",  +  2",  +  2i")  L(- H".  +  1^,  + D- 

634.  [4]  P  at  -  34".  Pass  a  plane  tangent  to  a  right  cone  whose  base  is  a  If' 
circle  lying  in  a  plane  perpendicular  to  the  ground  line  with  center  at 
R(_li''^_li", -11'')   and  whose  altitude  is  3|" : 

(a)  Through  a  point  A(- 2f ',  -  1",  z")  on  the  surface. 

(b)  Through  a  point  B(-  2f",  -  ^",  -  -^"),  not  on  the  surface. 

(c)  Parallel  to  line  S(- 3^",  -  T,  -  2")  T(- f",  +  f",  -  !")• 

635.  [2]  Pass  a  plane  tangent  to  a  right  cone  having  for  its  base  the  ellipse 
whose  major  axis  is  4i"  long- and  makes  +30°  with  the  ground  line,  and  whose 
minor  axis  is  2f"  long  and  intersects  the  major  axis  at  point  0(-  4|",  -  2|",  0"). 
Its  base  is  in  the  H  plane  and  its  altitude  4^". 

(a)  Through  point  D(- 3f",  y,  -  l^")  on  the  surface. 

(b)  Through  point  Q(-  6",  -  3f",  -  1|"),  off  the  surface. 

(c)  Parallel  to  line  R(-  4i",  -  f",  -  D  S(-  6",  -  If",  -  31"). 

4.— PLANES  TANGENT  TO  OBLIQUE  CONES. 

636.  [2]  Pass  a  plane  tangent  to  the  cone  whose  base  is  the  2-^"  circle  lying 
in  a  plane  parallel  to  H  with  center  at  A(- 2|",  -  3",  -  4^")  and  whose  vertex  is 

at  B(-6r,-n-r): 

(a)  Through  point  C(- oV',  -  If",  z)  on  the  surface. 

(b)  Through  point  D(- 3|",  -  1|",  -  3"),  not  on  the  surface. 

(c)  Parallel  to  line  E(-  2^",  -  2^",  -  3")  F(-  6^",  -  2|",  +  3"). 

637.  P  at  -  3^'.  Pass  a  plane  tangent  to  the  oblique  cone  whose  base  is  a  2|" 
circle  lying  in   P  with  center  at  L(0",  -  1/^",  -  2-|"),  and   whose  vertex   is   at 

M(-3r,-3J",-4f")- 

(a)  Through  point  N(- 1^",  y,  -  3^")  on  the  surface. 

(b)  Through  point  0(-^",  -  2",  -  1"). 

(c)  Parallel  to  line  R(-  f",  -  f",  -  i")  Q(-  3|",  -  Y',  -  ^D- 

638.  [2]  Pass  a  plane  tangent  to  the  cone  whose  base  is  the  ellipse  with  major 
axis  A(-6",  +  3",  0")  B(- U",  +  3",  0")  and  minor  axis  C(- 3f",  +  4f",  0") 
P(_3|'',  +  iy',  0")   and  whose  vertex  is  E(- 7i",  +  4-i",  +  4^"). 

(a)  Through  a  point  M  on  the  surface. 

(b)  Through  a  point  N  not  on  the  surface. 

(c)  Parallel  to  the  line  F(- 2f",  +  f",  +  1")  G(- 4i",  +  3^",  +  ^'0- 

639.  [2]  Pass  a  plane  tangent  to  the  oblique  cone  having  0(-  li",  +  4f ,  +  1^') 
for  its  apex  and  the  3f"  circle  lying  in  V  with  its  center  at  N(- 6J",  0",  +  3") 
for  a  base : 

(a)  Through  the  point  Q(- 2J",  +  2|",  z)  on  the  surface. 

(b)  Through  the  point  R(-  2",  +  2^",  +  2f ") . 

(c)  Parallel  to  the  line  S(-  4^,  +  -if",  +  3^")  T(-  7",  +  f",  +  3f"). 


43 

5.— PLANES  TANGENT  TO  HELICAL  CONVOLUTES. 

640.  [2]  A  convolute  is  generated  by  a  line  moving  tangent  to  the  helix  gen- 
erated by  the  point  M(-2^-",  +  Ig",  0'')  moving  in  a  negative  direction  about. line 
N(-4i^  +  ir,  +  2r)  O(-4J",  +  ir,0'')  with  a  pitch  of  2".  Pass  a  plane 
tangent  to  the  convolute : 

(a)  Through  point  Q(- 3f", -t- 3f",  z)  on  the  surface. 

(b)  Through  point  R(-7p,  + 3",  + f'),  not  on  the  surface. 

(c)  Parallel  to  line  S(- 2f' ,  +  f",  +  1")   T(+ 5f",  +  If ',  +  2-D. 

641.  [2]  G.  L.  :["  above  lower  border  of  working  space.  A  convolute  is  gen- 
erated by  a  line  moving  tangent  to  the  helix  generated  by  pt.  A(-  6f",  -  7f ,  0'') 
moving  in  a  positive  direction  about  line  B(-  5f ',  -  7f ',  0")  C(-  5f",  -  7f ',  +  4'') 
with  a  pitch  of  3|".     Pass  a  plane  tangent  to  the  convolute. 

(a)  Through  point  D(- 7",  -  of",  z)  on  the  surface. 

(b)  Through  point  E(- 31'',- 51'',  + li'O.  off  the  surface. 

(c)  Parallel  to  line  F(-5-J",- 91", +  21")   G(- 44-",  -  Sf",  +  f'). 

642.  [2]  A  convolute  is  generated  by  a  line  moving  tangent  to  the  helix  gen- 
erated by  pt.  A(- 6f", +  4|",  0")  moving  in  a  negative  direction  about  line 
J(-  5i",  +  43",  0")  K(-  Sf,  +  4r',  -u  4")  with  a  pitch  of  4^".  Pass  a  plane  tan- 
gent to  the  convolute : 

(a)  Through  point  L(- 6",  +  2|",  z)  on  the  surface. 

(b)  Through  point  M(-  3f",  ^  lf",  +  li"),  not  on  the  surface. 

(c)  Parallel   to   line   N(- 2",  +  If",  +  21")    0(- 4i".  +  H",  +  3|"). 

6.— PLANES  TANGENT  TO  HYPERBOLIC  PARABOLOIDS. 

643.  [2]  Pass  a  plane  T  tangent  to  the  hyperbolic  paraboloid  of  Prob.  585 
through  the  point  0(-4", +  iy,  z)  on  the  surface. 

644.  [1]  Pass  a  plane  S  tangent  to  the  hyperbolic  paraboloid  of  Prob.  587  at 
a  point  R  on  the  surface  whose  horizontal  proj.  is  at  the  point  (- 11",  +  2",  0"). 

645.  [2]  Assume  P  at  +  5".  Pass  a  plane  U  tangent  to  the  hyperbolic  para- 
boloid of  Prob.  589  through  the  point  Q  on  the  surface. 

646.  [1]  Pass  a  plane  T  tangent  to  the  hyperbolic  paraboloid  of  Prob.  590  at 
a  point  O  on  the  surface  whose  V  projection  is  at  the  point  (-  9|",  0",  -  2f"). 

647.  [2]  Assume  P  at  +  4|".  Pass  a  plane  R  tangent  to  the  hyperbolic  para- 
boloid of  Prob.  591  through  that  point  O  on  the  surface  which  is  verticallv  pro- 
jected at  (-9i",  0",-li"). 

648.  [2]  Pass  a  plane  S  tangent  to  the  hyperbolic  paraboloid  of  Prob.  592  at 
the  point  K(-  2§",  y,  -  2")  on  the  surface. 

649.  [2]  Pass  a  plane  R  tangent  to  the  hyperbolic  paraboloid  in  Prob.  584 
through  the  point  O  on  the  surface  whose  V  projection  is  at  (-  6f ,  0",  -  2f ). 

650.  [2]  The  right  lines  C(- 7",  -  2i",  -  3")  D(- 5i",  -  f'^  _  i^^)  and 
A(-2f", -1", -1")  B(-2-2". -1", -3-}")  are  directrices  of  a  Hyperbolic  Para- 
boloid whose  plane  directer  is  PI.  Find  a  plane  directer  of  the  2nd  generation, 
and  a  plane  R  tangent  to  the  surface  at  a  point  K(-  5",  y",  -  2^"). 

7.— PLANES  TANGENT  TO  HYPERBOLOIDS  OF  ONE  NAPPE. 

651.  [2]  Pass  a  plane  T  tangent  to  the  hyperboloid  of  Prob.  602  through  the 
point  0(- 4f ,  -  If",  z)  on  the  surface. 

652.  [2]  Pass  a  plane  T  tangent  to  the  hyperboloid  of  Prob.  603  through  the 
point  R  on  the  surface  whose  H  projection  is  at  the  point  (-  4f",  +  f",  0"). 

653.  [2]  P  at  +4-^".  Pass  a  plane  T  tangent  to  the  hyperboloid  of  Prob.  600 
through  the  point  T(- 8f ,  y,  -  2|")  on  the  surface. 


44 

8.— PLANES  TANGENT  TO  HELICOIDS. 

65i.   [2]    Pass  a  plane  tangent  to  the  helicoid  of  Prob.  607, 

(a)  Through  the  point  0(- 3^'',  y'',  z")  on  the  hehcal  directrix. 

(b)  Through  the  point  R(- 3^'', -1^'',  y'')  on  the  surface. 

655.  [1]  G.  L.  parallel  to  short  edge  of  sheet  and  2"  below  middle  of  sheet. 
Pass  a  plane  tangent  to  the  helicoid  of  Prob.  610, 

(a)  Through  the  point  A(- 6|",  y",  z'')on  the  helical  directrix. 

(b)  Through  the  point  B  on  the  surface  which  is  horizontally  projected 
at  (-5'',-2r,0''). 

656.  [2]   Find  the  traces  of  a  plane  tangent  to  the  helicoid  of  Prob.  611. 

(a)  Through  the  point  A(- 4f'',  y'',  z'')  on  the  helical  directrix. 

(b)  Through  the  point  B(- 3",  +  ly,  z")   on  the  surface. 

657.  [2]    Pass  a  plane  tangent  to  the  right  helicoid  of  Prob.  608. 

(a)  Through  the  point  R(- 4i",  y'',  z")  on  the  helical  directrix. 

(b)  Through  the  point  S(-  5^",  y",  +  3'')  on  the  surface. 

658.  [1]  Find  the  traces  of  a  plane  tangent  to  a  helicoid  of  3"  pitch,  whose  axis 
is  C(-9i", +  2^",  0")  D(-9^'', +  2|", +  4"),  initial  position  of  generatrix 
£(_7i''^  +  2^'',  6")  F(-9-i-", +  2^', +  2^'')  which  moves  so  that  its  H  projection 
rotates  clockwise. 

(a)  Through  the  point  A(-  lOf'',  y ,  z")  on  the  helical  directrix. 

(b)  Through    the    point    B    on    the    surface    which    is    projected    at 

(-8r,+3r,z'0- 

9.— PLANES  TANGENT  TO  SPHERES. 

659.  [2]  A  sphere  of  4"  diameter  is  located  in  the  first  quadrant  with  its  cen- 
ter at  the  point  0(-  h\" ,  +  2^",  +  2^").  Find  a  plane  T,  tangent  to  the  sphere  at 
that  point  P  on  the  bottom  portion  of  the  surface  whose  H  projection   is  at 

660.  [2]   Find  a  plane  S  tangent  to  the  above  sphere  through  the  right  line 

A(-3-,  +  2r,+ir)  B(-r,+r,  +  3r)- 

661.  [2]  A  sphere  of  o"  diameter  rests  upon  a  horizontal  plane  in  the  3rd 
quadrant  at  the  point  M(- 3J",  -  2",  -  3fO-  Fintl  a  plane  T  tangent  to  the 
sphere  at  a  point  P  on  the  upper  portion  of  the  surface  whose  H  projection  is 

(-2f'',-l'',0'0- 

662.  [2]   Find  a  plane  U  tangent  to  the  above  sphere  through  the  right  line 

c(- 4r, -  5-,  +  r )  D(-  7i-, - \'\ -  3r ). 

663.  [2]  P  at  -3i".  A  sphere  of  2^"  diameter  is  located  in  the  third  quad- 
rant with  its  center  at  the  point  0(-  \\" ,  -  \\" ,  -  \%").  Find  the  tangent  plane 
T   to  the   sphere   at   the   point   A    on    the    surface   whose    H    projection    is    at 

10.— PLANES  TANGENT  TO  ELLIPSOIDS. 

664.  [2]  An  ellipsoid  of  revolution  is  formed  in  the  third  quadrant  by  the  rota- 
tion about  its  vertical  major  axis  of  an  ellipse  whose  axes  are  respectively  3'' 
and  2".     Find  a  plane  tangent  to  the  surface  at  some  convenient  point  R. 

665.  [2]  An  ellipsoid  of  revolution  is  formed  in  the  3rd  quadrant  by  the  rota- 
tion about  its  vertical  minor  axis,  of  an  ellipse  whose  axes  are  3"  and  If"  respec- 
tively.    Find  a  plane  tangent  to  the  surface  at  a  convenient  point  A. 

666.  [2]  Find  a  plane  tangent  to  the  above  ellipsoid  through  a  convenient 
line  EC. 

667.  [2]  An  ellipsoid  of  revolution  is  formed  in  the  3rd  quadrant  by  the  rota- 
tion about  its  vertical  major  axis  of  an  ellipse  whose  axes  are  4''  and  2|''  respec- 
tively.   Find  a  plane  tangent  to  the  surface  at  a  point  \"  from  the  top. 

668.  [2]  An  ellipsoid  of  revolution,  axes  6''  and  3"  respectively,  has  its  center 
at  the  point  0(- 41:'',  +  2",  0"),  and  its  long  axis  perpendicular  to  H.     Find  a 


45 

plane  tangent  to  the  ellipsoid  at  a  point  B  on  the  surface  horizontally  projected 
at  point  {-3V',  +  1Y',0''). 

669.  [21  Find  a  plane  N,  tangent  to  the  above  ellipsoid  and  through  the  line 
A(-  6f ^  +  If",  +  r )   B(-  5-,  +  4r,  +  3^') . 

11.— PLANES  TANGENT  TO  HYPERBOLOIDS  AND  PARABOLOIDS 

OF  REVOLUTION. 

670.  [3]  A  Hyperbola,  in  a  plane  parallel  to  V,  has  its  foci  at  the  points 
F(-5i",-2r'.-2i'')  and  G(- H'',-2V',-2V')  and  its  vertices  at  the  points 
V(-  4|",  -  2V',  -  2V'),  W(-  3^',  -  2f ',  -  2i")-  This  Hyperbola  spins  about  its 
vertical  conjugate  axis  and  generates  a  Hyperboloid  of  Revolution.  Find  a  plane 
T,  tangent  to  the  surface  at  the  point  K  on  the  surface  which  is  projected  at 

(-3r,y,-i"). 

671.  [3]  The  line  A(- 5f ',  -  3^,  -  i'')  B{- r',-2Y\- V')  is  the  directrix 
and  the  point  F(- 3^",  -  3|'',  -  1")  the  focus  of  a  parabola  which  is  the  genera- 
trix of  a  paraboloid  of  revolution  formed  by  rotating  the  parabola  about  its  axis. 
Find  a  plane  T  tangent  to  the  surface  at  a  point  K  on  the  surface. 

673.  [3]  Pass  a  plane  T  tangent  to  the  above  paraboloid  through  the  line 
K(-  6^' -  t",  -  4i")   M(-  4i",  -  Sr,  -  D. 

673.  [3]  A  hyperbola,  in  a  plane  parallel  to  V,  has  its  foci  at  the  points 
F(-4i",-3|",-lf'0  and  {^{-A^,",-2\",-'i\"),  and  its  vertices  at  the  points 
V(-4^':, -2i",-  3")  and  W(- 4|^  -  21'',  -  3").  This  hyperbola  spins  about  its 
vertical  transverse  axis  and  generates  a  Hyperboloid  of  Revolution.  Find  a 
plane  S  tangent  to  the  surface  through  the  point  O  on  the  surface  whose  V  pro- 
jection is  at  the  point  (- 4|",  0",  -  3]''). 

674.  [3]  P  at  +3".  The  line  A(- 81'',  -  2'',  -  4")  B(- 8^",  -  2",  0")  is  the 
directrix  and  the  point  F(- 7^",  -  2",  -  2")  the  focus  of  a  parabola  which  is  the 
generatrix  of  a  paraboloid  of  revolution  with  AB  as  its  axis.  Find  the  plane  T 
tangent  to  the  surface  at  a  point  P  on  the  surface  which  is  horizontally  projected 
at  p(-8",-l|",  0"). 

675.  [3]    P  at  +  3".     Find  a  plane  U  tangent  to  the  above  surface  through  line 

K(-3r,--3r,  +  r)  N(-5r/-r,  +  ir)- 

12.— PLANES  TANGENT  TO  TORI. 

676.  [3]  A  Torus  is  formed  by  the  revolution  of  a  circle  of  W  diameter 
about  the  axis  A(  -  4  J",  -!  3f' ,  +  4")  B(-  4-3",  +  3^",  0").  The  circle  In  its  initial 
position  is  parallel  to  V  with  its  center  at  the  point  0(-  6|",  +  3f",  +  If").  Find 
a  plane  T  tangent  to  the  surface  through  the  pt.  M  on  the  bottom  of  the  surface 
horizontally  projected  at  M(- 3|",  +  4",  0"). 

677.  [3]  A  Torus  is  formed  bv  the  revolution  of  a  circle  of  If"  diameter  about 
the  axis  A(-  3^",  -  3f",  -  3")  b'(-  3^",  -  3f",  0").  The  circle  in  its  initial  posi- 
tion is  parallel  to  V  with  its  center  at  the  point  0(- 4f",  -  3f",  -  If").  Find  a 
plane     U     tangent     to     the     surface     through     the     line     M(- 4|",  0",  -  3f") 

N(:-6r,-ir,-r)- 

678.  [1]  A  Torus  is  formed  by  the  revolution  of  a  circle  of  3"  diameter,  about 
the  axis  A(-  6^",  -  1-J",  -  4")  B(-  6|",  -  U",  0").  The  circle  in  its  initial  posi- 
tion is  parallel  to  V  with  its  center  at  the  point  0(- 7f",  -  1|",  -  2").  Find  a 
plane  S  tangent  to  the  surface  at  a  point  on  the  lower  portion  of  the  surface, 
whose  H  projection  is  at  (-  8|",  -  f ",  0"). 

679.  {1]  Find  a  plane  R  tangent  to  the  surface  of  Prob.  678,  through  the  line 
K(-8f",-31",-^")  L(-10f",-f",-3f"). 

680.  [1]  (Ground  line  parallel  to  short  edge  of  sheet.)  A  Torus  is  formed 
by  the  revolution  of  a  circle  of  3^"  diameter  about  the  axis  A(- 3f",  -  5",  -  4y ) 
B(-  3f",  0",  -  4^").  The  circle  in  its  initial  position  is  parallel  to  H  with  its  cen- 
ter at  the  point  0(- 6|",  -  3",  -  4^").  Find  a  plane  Q  tangent  to  the  surface 
through  the  point  R(-  6^",  y",  -  6")  on  the  upper  part  of  the  surface. 


TO  FIND  POINTS  WHERE  LINES  PIERCE  SURFACES. 

Right  Prisms  and  Pyramids. 

685.  [2]   Assume  P  at  -f  9.     Find  the  points  where  line  A(- 12",  -  1^",  -  2-|") 
B(-15f", -2-^", -f )  pierces  the  right  prism  of  Prob.  725. 

686.  [2]   Assume    P    at    h  0.      Find    where    the    line    C(- 12i",  - 1^",  -  SJ") 
D(-  15",  -  If,  -  If")  pierces  the  right  prism  of  Prob.  726. 

687.  [2]   Find  the  points  where  Hne  E(-  2f ",  +  2|",  + 1")  F(-  5^",  +  1",  +  If) 
pierces  the  right  pyramid  of  Prob.  730. 

688.  [2]   Find    where    the    line    G(- 2f",  +  If",  +  If )     H(- of ,  +  3",  +  .}") 
cuts  the  right  pyramid  of  Prob.  730. 

Oblique  Prisms  and  Pyramids. 

689.  [2]    Find  where  the  line  J(-  3^",  +  f",  +  3")  K(-  5f",  f  2f".  +  ^")  pierces 
the  oblique  prism  of  Prob.  727. 

690.  [2]   Find  the  pts.  where  line  L(- 3f ,  +  3^",  +  1")   KC- 6f ,  -  f ,  +  If") 
pierces  the  oblique  prism  of  Prob.  727. 

691.  [2]   Assume    P    at     1-9.      Find    where    the    line    A(- llf",  -  ^",  -  3|") 
B(-  13^",  -  2f",  -  1-^")  pierces  the  oblique  pyramid  of  Prob.  731. 

692.  [2]   Assume  P  at  +9.     Find  the  points  where  line  C(- llf ,  +  If ,  +  |") 
D(-13f , +  2", +  3")  pierces  the  oblique  pyramid  of  Prob.  732. 

Right  Cylinders  and  Cones. 

693.  [2]   Assume  P  at  f  9.     Find  the  points  where  the  right  cylinder  of  Prob. 
734  is  pierced  by  the  line  A(- llf ,  +  3f ,  +  3|")  B(- 15^",  +  f ,  +  If ). 

694. '[21   Find    where   line.  C(- 2",  -  1",  -  3f  )    D(- 5f ,  -  1^',  -  f  )    pierces 
the  right  cylinder  of  Prob.  736. 

695.  [2]   P  at  -  3f .  Find  where  line  E(-  V',  -  f",  -  3^")  F(-  3^",  -  3^",  -  i") 
pierces  the  right  cylinder  of  Prob.  738. 

696.  [2]   Assume    P    at    -i  8i".      Find    where    the    line    G(- llf ,  +  1",  +  f") 
K(-15f  ,  +  3",-:-2f )  pierces  the  right  cone  of  Prob.  747. 

697.  [2]   Assume  P  at  +  8J/'.    Find  where  the  right  cone  of  Prob.  748  is  pierced 
by  the  line  A(-  11",  +  f ,  -  V')  B(-  14^",  +  4^",  -^  2f ). 

698.  [2]   P  at  -  2".    Find  where  line  C(-  f ,-  3f ,-  2f")  D(-  4f",+  1",-  3f ) 
pierces  the  right  cone  of  Prob.  752. 

Oblique  CyHnders  and  Cones. 

699.  [2]   Assume      P     at  +  S".        Find      where      line      E(- lOJ",  -  i",  - 1") 
F(-13f , -2f , -44")  pierces  the  oblique  cylinder  of  Prob.  740. 

700.  [2]    P  at  0".     Find  the  points  where  the  oblique  cylinder  of  Prob.  741  is 
pierced  by  line  A(- 2^",  -  1",  -  4f )  B(- 6f ,  -  4",  -  1"). 

701.  [2]   Assume  P  at  +  9".     Find  the  points  in  which  the  oblique  cylinder  of 
Prob.  743  is  pierced  by  line  C(- 11",  +  4f , +  3")  D(- 15",  +  If ,  +  f )'. 

702.  [2]   Assume     P     at     +8".       Find     where     line     A(- 9f ,  +  f ,  - 1^") 
B(-12f ,  -  2f , -3f )  pierces  the  oblique  cone  of  Prob.  763. 

703.  [2]  Assume     P     at    +  8f .       Find     where     line     C(- lOf ,  -  1^",  -  4") 
D(-14f  ,  +  f , -If )  pierces  the  oblique  cone  of  Prob.  764. 

704.  [2]   Assume  P  at  +8^".     Find  the  points  in  which  the  oblique  cone  of 
Prob.  765  is  pierced  by  the  line  E(-  lOf",  -  f ,  -  3")  F(-  13|",  -  3",  -  4"). 


47 

Warped  Surfaces. 

705.  [2]  Assume   P   at  +  U'\     Find   where   the   line   A(- 5^",  -  Sf", -4f'0 
B(-10f' , -§",  -2'')  pierces  the  hyperbolic  paraboloid  of  Prob.  795. 

706.  [2]   Assume   P  at  +5-]:''.     Find  where  the  Hne   C(- 10|",  -  3f ',  -  4i") 
D(- 12-1", -4", -3'')  pierces  the  hyperbolic  paraboloid  of  Prob.  793. 

707.  [2]   Find    where    the    hvperboloid    of    Prob.    806    is    pierced    by    line 

E(-  6r,  -  v,  -  5'')  F(-  2f^  -  ir, + 1|"). 

708.  [2]   Assume  P  at  H-  7".    Find  where  the  hyperboloid  of  Prob.  805  is  pierced 
by  the  line  A(-  8^',  +  ^'',  +  f )  B(-  13|",  +  f",  +  2f")- 

709.  [2]   Assume     P     at     +74".        Find     where     line     C(- 9^",  + 1",  -  i") 
D(-  14i",  +  3|",  ^  4")  pierces  the  helicoid  of  Prob.  811. 

710.  [2]   Find  where  the  helicoid  of  Prob.   812   is  pierced  by   the   right   line 

A(- ir.. - r, - 1")  B(- 6r, - sr, - sf). 

Double  Curved  Surfaces. 

711.  [2]   Find  where  line  A(- 3",  +  1",  +  l^")    B(- 6i",  +  2^',  +  4^")    pierces 
the  sphere  of  Prob.  767. 

712.  [2]   Find  where  the   sphere  in   Prob.   768   is  pierced  bv   the  right   line 

c(-r,+r,  +  ir)  d(-5",-4",-3"). 

713.  [2]   Find  where  the  ellipsoid  of  Prob.  770  is  pierced  by  a  line  conveniently 
assumed. 

714.  [2]    Find  the  points  in  which  the  ellipsoid  of  Prob.  773  is  pierced  by  the 
line  E(-  ir,  +  3-1",  +  3|")  F(-  61",  +  2-J",  +  f"). 

715.  [2]   Find   where   line   A(- 2i",  -  T,  -  3|")    B(- 5^",  -  2^,  -  |")    pierces 
the  paraboloid  of  Prob.  775. 

716.  [2]   Find   where   the   torus   of    Prob.    779    is   pierced   by    the   right   line 
C(-2^",  +  4|",  +  li")   D(-7",  +  |",  +  2f"). 

717.  [2]    Find  the  points   in  which  the  given   right  line  K(- 1|",  -  J",  -  J") 
M(-  6f",  -  4f",  -  2f")  pierces  the  torus  of  Prob.  780. 


I.     INTERSECTIONS  OF  SURFACES  AND  PLANES. 


1.— PRISMS  CUT  BY  PLANES,  AND  DEVELOPMENT. 

725.  [1]  The  line  M(- 14'',  -  If^ -4'^  N(- 14",  -  If ,  0")  is  the  axis  of  a 
regular  hexagonal  prism,  whose  upper  base  is  in  H  with  center  at  the  point  N. 
Each  side  of  the  base  is  ly,  and  two  sides  are  parallel  to  the  ground  line.  The 
prism  is  cut  by  a  plane  T(- 11|-",  f  120°,  -  135°).  Find  the  intersection  of  the 
plane  and  the  prism,  the  actual  size  of  the  section  when  swung  into  H,  and  the 
development  (pattern)  of  that  part  of  the  prism  between  the  planes  H  and  T. 
Develop  prism  base  along  the  ground  line,  beginning  at  a  point  (-9")  with  the 
shortest  element  and  developing  to  the  right  and  below  the  ground  line.  (State 
problem  in  upper  right  portion  of  sheet.) 

726.  [1]  The  line  M(- 13|'',  -  1^',  -  4")  N(- 13|",  -  1^',  0")  is  the  axis  of 
a  regular  square  prism,  whose  base  is  in  a  plane  4''  below  H  and  parallel  to  H. 
The  center  of  the  square  base  is  at  the  point  M,  each  of  the  sides  is  1|"  long,  and 
two  of  them  when  extended  make  angles  of  +60°  with  the  ground  line.  The 
prism  is  cut  off  by  the  plane  T(- 9f' ,  +  135°,  -  150°).  Find  the  intersection  of 
the  plane  and  the  prism,  the  actual  size  of  the  section  when  swung  into  V,  and 
the  development  (pattern)  of  the  prism,  between  T  and  the  plane  4"  below  H. 
Develop  the  prism  base  on  the  line  4"  below  the  ground  line,  beginning  with  the 
shortest  edge  at  a  point  4|"  to  the  right  of  m'  and  developing  to  the  right  and 
above  the  base  line.     (State  problem  in  upper  right  hand  portion  of  sheet.) 

727.  [1]  The  line  M(- 6",  +  H".  +  3|'')  N(- 3|",  +  2",  0")  is  the  axis  of  a 
triangular  prism  whose  base  is  an  equilateral  triangle  in  H.  This  triangle  is  in- 
scribed in  a  circle  of  2V'  diameter  and  center  at  N,  with  its  side  nearest  to  V 
parallel  to  the  ground  line.  The  prism  is  cut  by  the  plane  T(-  7^",  -  45°,  +  30°). 
Find  the  intersection  of  the  plane  and  prism,  the  true  size  of  the  section  when 
swung  into  H,  and  the  development  (pattern)  of  that  part  of  the  prism  between 
T  and  H.  (Develop  the  prism  in  the  upper  left  hand  portion  of  the  sheet,  and 
state  the  problem  in  the  lower  left  hand  portion.) 

728.  [2]  Find  the  intersection  of  that  portion  of  a  plane  included  between 
the  right  lines  M(-  6i",  +  T,  +  D  N(-  4^'',  -  t",  +  3f'0  and  0(-  5^",  +  2^",  O'O 
P(-3J:",  +  f' ,  +  4")  with  the  prism  made  up  of  lines  A(- ^'',  + V',  +  2^'') 
E(-  IV,  +  V,  +  3D  and  the  lines  B(-  5f',+  1",+  f")  F  and  C(-  4|",+"2^",+  f") 
D,  drawn  parallel  to  AE  through  the  points  B  and  C. 

729.  [2]  The  line  M(- 5",  0'',  -  2")  N(- 24'',  -  3|",  -  1^")  is  the  axis  of  a 
square  prism  whose  base  is  in  V,  center  at  point  M,  with  2  of  its  3"  sides  parallel 
to  the  G.  L.  Find  the  intersection  of  this  prism  with  an  opaque  plane 
T(- 1",  +  150°,  - 135°),  indicating  visible  and  invisible  edges. 

2.— PYRAMIDS  CUT  BY  PLANES  AND  DEVELOPMENT. 

730.  [1]  The  line  M(- 4",  i- 2",  +  3^")  N(- 4",  +  2",  0")  is  the  axis  of  a  reg- 
ular hexagonal  pyramid.  Its  base  in  H  has  sides  1|"  long,  two  of  these  sides 
being  parallel  to  V.  The  point  M  is  the  vertex.  The  pyramid  is  cut  by  the  plane 
TC- 8^", -45°,  + 30°).  Find  the  intersection  of  the  plane  and  the  pyramid,  the 
actual  size  of  this  section  when  swung  into  H,  and  the  development   (pattern) 


49 

of  the  pyramid,  showing  the  intersection.     (Develop  pyramid  in  upper  left  hand 
portion  of  the  sheet,  and  state  problem  in  lower  left  hand  portion.) 

731.  [1]  The  point  M(- lU'',  -  J",  -  1")  is  the  vertex  of  a  pyramid  which 
is  cut  by  a  plane  T(-  16",-!-  30°,  -  45°).  The  base  of  the  pyramid  is  a  2"  square 
in  a  plane  4"  below  H  and  parallel  to  H.  The  center  of  the  square  is  the  point 
0(-  12V',  -  lY',  -  4")  and  two  of  its  sides  are  parallel  to  the  H  trace  of  plane  T. 
Find  the  intersection  of  the  plane  and  the  pyramid,  the  actual  size  of  this  section 
when  swung  into  H,  and  the  development  (pattern)  of  the  pyramid  showing 
the  intersection.  (Develop  pyramid  in  upper  right  hand  portion  of  the  sheet,  and 
state  problem  in  lower  right  hand  portion.) 

732.  [1]  The  point  0(- 12",  -  3",  -  U")  is  the  vertex  of  a  pyramid  which  is 
cut  by  a  plane  T(- 9^",  +  150°,  -  120°).  The  base  of  the  pyramid  in  V  is  an 
equilateral  triangle  inscribed  in  a  circle  of  2"  diameter,  whose  center  is  at  the  point 
M(-13r',  0", -1-r').  The  upper  side  of  this  base  is  parallel  to  H.  Find  the 
intersection  of  the  plane  and  the  pyramid,  the  actual  size  and  shape  of  the  sec- 
tion when  swung  into  V,  and  the  development  (pattern)  of  the  pyramid,  show- 
ing the  intersection.  (Develop  the  pyramid  in  the  upper  right  hand  portion  of 
the  sheet,  and  state  the  problem  in  the  lower  right  hand  portion.) 

733.  []]  P  at  -  7".  Find  3  projections  and  the  intersection  of  the  right  square 
pyramid,  having  an  altitude  of  5"  and  having  line  A(- 5^",  -  f",  z) 
B(-  7",  -  21",  z)  in  plane  T(-  6^",  4  120°,  -  67J-°)  for  one  side  of  its  base  which 
is  in  plane  T  and  does  not  cross  its  trace,  with  the  opaque  parallelogram 
E(-3",-U",-3")  F(-5",-l", -f )  G(-6",-4i", -li")   H. 

3.— RIGHT  CYLINDERS  CUT  BY  PLANES. 

Curve  of  intersection,  true  size,  and  development. 

734.  [1]  The  line  M(- 14",  +  2",  0")  N(- 14", -h  2",  +  4")  is  the  axis  of  a 
right  cylinder,  whose  circular  base  of  3"  diameter  is  in  H.  The  cylinder  is  cut 
by  the  plane  T(- 11^",  -  112-|°,  h  135°).  Find  the  curve  of  the  intersection  of 
the  plane  and  the  cylinder,  the  true  size  of  this  curve  when  swung  into  H,  and 
the  development  of  that  part  of  the  cylinder  between  H  and  T.  Develop  the 
cylinder  base  along  the  ground  line,  beginning  at  a  point  (-  10^')  with  the 
shortest  cylinder  element,  and  developing  to  the  right  and  above  the  ground  line. 
(State  the  problem  in  lower  right  hand  portion  of  sheet.) 

735.  [1]  The  line  M(- 14",  -  2",  -  4")  N(- 1-4",  -  2",  0")  is  the  axis  of  a 
right  cylinder  whose  circular  base  of  3"  diameter  is  in  a  plane  parallel  to  H  and 
4"  below  H.  The  cylinder  is  cut  by  the  plane  T(- 15",  +  120°,  -  45°).  Find 
the  curve  of  intersection  of  the  plane  and  the  cylinder,  the  true  size  of  this  curve 
when  swung  into  H,  and  the  development  of  that  part  of  the  cylinder  between  T 
and  the  horizontal  plane  4"  below  H.  Develop  the  cylinder  base  on  a  line  parallel 
to  the  ground  line  and  4"  below  it,  beginning  at  a  point  10|"  to  the  left  of  the 
right  hand  border  line.  Begin  with  the  shortest  cylinder  element  and  develop 
to  the  right  and  above  the  base  line.  (State  the  problem  in  upper  right  hand 
portion  of  sheet.) 

736.  [1]  The  line  M(- 4",  -  IJ",  -  4")  N(- 4",  -  IJ",  0")  is  the  axis  of  a 
right  cylinder  whose  upper  base  is  an  ellipse  in  H.  The  major  and  minor  axes 
of  the  ellipse  intersect  at  the  point  N  and  are  respectively  2|"  and  1|"  in  length. 
The  major  axis  is  parallel  to  the  H  trace  of  the  plane  S'(- ^i". +  30°,  -  22^°) 
which  cuts  the  cylinder.  Find  the  curve  of  intersection  of  the  plane  and  the 
cylinder,  the  true  size  of  the  curve  when  swung  into  H,  and  the  development  of 
that  part  of  the  cylinder  between  H  and  T.  Develop  the  cylinder  base  along  the 
ground  line,  beginning  at  a  point  (-  9f ")  with  the  shortest  element  and  develop 


50 

to  the  left  and  below  the  ground  line.     (State  problem  in  upper  left  hand  portion 
of  sheet.) 

737.  [1]  The  line  M(- 3",  -  4'',  -  2")  N(_  3"^  o",  -  2'0  is  the  axis  of  a  right 
cylinder,  whose  circular  base  of  2''  diameter  is  in  V.  The  cylinder  is  cut  by  a 
plane  T(- 6'', +  45°,  -  60°).  Find  the  curve  of  intersection  of  the  plane  and  the 
cylinder,  the  true  size  of  this  curve,  when  swung  into  V,  and  the  development  of 
that  part  of  the  cylinder  between  V  and  T.  Develop  the  cylinder  base  along  the 
ground  line,  beginning  at  a  point  (-SI")  with  the  shortest  cylinder  element  and 
developing  to  the  left  and  above  the  ground  line.  (State  problem  in  lower  left 
Iiand  portion  of  sheet.) 

738.  [1]  Profile  plane  at  -10''.  The  line  M(- 5'',  -  2",  -  2'')  N(0'',  -  2",- 2'') 
is  the  axis  of  a  right  cylinder,  whose  circular  base  of  2J"  diameter  is  in  P.  The 
cylinder  is  cut  by  a  plane  T(-  %'',  +  120°,-  120°).  Find  the  curve  of  intersection 
of  the  plane  and  the  cylinder,  the  true  size  of  the  curve  when  swung  into  H,  and 
the  development  of  that  part  of  the  cylinder  between  P  and  T.  Develop  the  cylin- 
der base  along  the  ground  line,  beginning  at  a  point  (-8^'')  with  the  shortest 
element,  and  developing  to  the  right  and  above  the  ground  line.  (State  problem 
in  lower  rieht  hand  portion  of  sheet.) 


4.— OBLIQUE  CYLINDERS  CUT  BY  RIGHT  SECTION  PLANES  AND 

DEVELOPMENT. 

739.  [1]  (Place  ground  line  4i"  below  top  edge  of  sheet.)  The  point 
N(- 12'',  +  2|:",  0")  is  the  center  of  the  circular  base  in  H  of  an  oblique  cylinder 
in  the  first  quadrant.  The  diameter  of  this  base  is  3".  The  elements  of  the  cylin- 
der are  such  that  their  PI  and  V  projections  when  extended  makes  angles  of  -  30° 
and  4-60°  respectively  with  the  ground  line.  The  cylinder  is  cut  by  a  right  section 
plane  R  whose  traces  meet  the  ground  line  at  (- 7^",  0",  0").  Find  the  H  and 
V  projection  of  the  curve  of  intersection  of  plane  and  cylinder,  the  true  size  of 
the  right  section  when  swung  into  H,  and  the  development  of  that  portion  of  the 
cylinder  between  H  and  T.  (Develop  cylinder  in  upper  right  hand  portion  of  the 
sheet,  and  state  problem  in  lower  right  hand  portion.) 

740.  [1]  The  point  N(-  13",  0",  -  1|")  is  the  point  of  intersection  of  the  major 
and  minor  axes  of  the  elliptical  base  in  \  of  an  oblique  cylinder  in  the  third  quad- 
rant. These  axes  are  4"  and  2"  in  length  respectively  and  when  extended  make 
angles  of  -  30°  and  -  120°  with  the  ground  line.  The  horizontal  and  vertical  pro- 
jections of  the  elements  of  the  cylinder  make  angles  of  +  45°  and  -  30°  with  the 
ground  line  respectively.  The  cylinder  is  cut  by  a  right'  section  plane  R,  whose 
traces  intersect  the  ground  line  at  (-  9",  0",  0").  Find  the  H  and  V  projections 
of  the  curve  of  intersection  of  plane  and  cylinder,  the  true  size  of  the  right  sec- 
tion when  swung  into  V,  and  the  development  of  that  portion  of  the  cylinder  be- 
tween V  and  T.  (Develop  cylinder  in  upper  right  hand  portion  of  sheet.  State 
problem  in  lower  right  hand  portion.) 

741.  [1]  P  at  -  8|".  The  point  N(-  5i",  -  1^",  -  5")  is  the  center  of  the  cir- 
cular base  of  3"  diameter  of  an  oblique  cylinder.  This  base  is  in  a  horizontal 
plane  5"  below  PI  and  the  cylinder  stands  in  the  third  quadrant.  Its  elements  are 
such  that  their  H  and  V  projections  make  angles  of  +  30°  and  -  135°  respectively 
•with  the  ground  line.  The  cylinder  is  cut  by  a  right  section  plane  R  whose  traces 
meet  the  ground  line  at  (-4",  0",  0").  Find  the  H,  V  and  P  projections  of  the 
cylinder  and  curve  of  intersection  of  the  cylinder  and  plane,  the  true  size  of  this 


section  when  swung  into  H,  and  the  development  of  that  part  of  the  cylinder 
between  its  base  and  R.  (Develop  cylinder  in  upper  right  hand  portion  of  sheet. 
State  problem  in  lower  right  hand  portion.) 


5.— OBLIQUE  CYLINDERS  CUT  BY  OBLIQUE  PLANES  AND 

DEVELOPMENT. 

743.  [1]  The  line  M(- 14",  +  l^",  0'')  N(- IQi",  +  3^",  +  3i'0  is  the  axis  of 
an  oblique  cylinder,  whose  right  section  is  a  circle  of  2"  diameter.  The  cylinder 
is  cut  by  a  plane  T(-  (i",  -  13.5°,  +  135°).  Find  the  base  of  the  cylinder  in  H, 
the  curve  of  intersection  of  the  plane  T  and  the  cylinder,  and  the  development  of 
that  part  of  the  cylinder  between  H  and  the  plane  T.  ( Develop  cylinder  in  upper 
right  hand  portion  of  sheet;  state  problem  in  lower  right  hand  portion.) 

743.  [1]  The  point  N(- 12'',  +  If",  0")  is  the  center  of  the  circular  base  in 
H  of  an  oblic|ue  cylinder.  The  diameter  of  this  base  is  3"  and  the  cylinder  stands 
in  the  first  quadrant.  The  H  and  V  projections  of  the  cylinder  elements  make 
angles  of  -  150°  and  +  120°  respectively  with  the  ground  line.  The  cylinder  is 
cut  by  two  planes,  one  the  plane  T(- 15^",  -  90°,  +  45°)  and  the  other  a  right 
section  plane  R  through  the  same  point  in  the  ground  line.  Find  the  projection 
of  the  curves  of  intersection  of  the  planes  and  the  cylinder  and  the  development 
of  that  portion  of  the  cylinder  between  T  and  R.  (Develop  cylinder  in  the  upper 
right  hand  portion  of  the  sheet;  state  the  problem  in  lower  rig'ht  hand  portion.) 

744.  [1]  The  point  N(- 13",  -  1^",  -  4")  is  the  center  of  the  circular  base  of 
2"  diameter  of  an  oblique  cylinder.  This  base  is  in  a  horizontal  plane  4'"  below 
H  and  the  cylinder  stands  in  the  third  quadrant.  The  H  and  V  projections  of  the 
cylinder  elements  make  angles  of  +45°  and  -120°  with  the  ground  line  respec- 
tively. The  cylinder  is  cut  by  a  plane  T(- 15",  +  90°,  -  45°)  and  by  a  right  sec- 
tion plane  R(- 13", +■  135°,  -  30°).  Find  the  projection  of  the  curves  of  inter- 
section of  the  planes  and  the  cylinder,  and  the  development  of  that  portion  of  the 
cylinder  between  T  and  R.  (Develop  cylinder  in  upper  right  hand  portion  of 
sheet;  state  problem  in  lower  right  hand  portion.) 

745.  [1]  The  point  N(- 11",  -  l-J",  0")  is  the  point  of  intersection  of  the  axes 
of  the  elliptical  base  in  H  of  an  oblique  cylinder  which  is  located  in  the  third  quad- 
rant. The  major  and  minor  axes  of  the  top  base  are  respectively  2"  and  1^"  in 
length,  the  former  being  parallel  to  the  ground  line.  The  H  and  V  projections  of 
the  elements  of  the  cylinder  make  angles  of  +  135°  and  -  120°  respectively  with 
the  ground  line.  The  cylinder  is  cut  by  a  plane  T(-l()",  +  G0°,  -  45°)  and  by  a 
right  section  plane  through  a  point  (-  14|-",  0",  0")  in  the  ground  line.  Find  the 
curves  of  intersection  of  these  planes  and  the  cylinder,  and  the  development  of 
that  part  of  the  cylinder  between  T  and  H.  (Develop  cylinder  in  upper  right 
hand  portion  of  sheet,  stating  problem  in  lower  right  hand  portion.) 

746.  [1]  P  at  -9".  llie  line  M(- 7",  -  3r',  -  3-^")  N(0",  -  1]",  -  1^")  is 
the  axis  of  an  oblique  cylinder  situated  in  the  third  quadrant,  whose  base  in  P 
is  a  circle  of  2"  diameter.  This  cylinder  is  cut  by  the  plane  T(-  7",  +  60°,  -  60°) 
and  by  a  right  section  plane  R  through  a  point  (-  3^',  0",  0")  on  the  ground  line. 
Find  the  curves  of  intersection  between  these  planes  and  the  cylinder,  and  the 
development  of  that  part  of  .the  cylinder  between  T  and  P.  (Develop  cylinder 
in  upper  right  hand  portion  of  sheet,  state  problem  in  lower  right  hand  portion.) 


52  . 

6.— RIGHT  CONES  CUT  BY  PLANES  AND  DEVELOPMENT. 

747.  [1]  The  line  M(- 12'^  +  2^,  +  ^D  N(- 12'',  +  2^",  0")  is  the  axis  of  a 
right  circular  cone  in  the  first  quadrant,  whose  vertex  is  at  M  and  whose  base  of 
4^"  diameter  is  in  H  with  its  center  at  N.  This  cone  is  cut  by  a  plane 
T(- 9",  -  90°,  f  150°).  Find  the  H  and  V  projections  of  the  curve  of  intersec- 
tion of  the  plane  and  cone,  the  true  size  of  this  curve,  and  the  development  of  the 
cone,  showing  the  intersection. 

748.  [1]  The  line  M(- 13^",  +  2^",  +  Sf")  N(- 13^",  +  2^",  0")  is  the  axis  of 
a  right  circular  cone,  whose  vertex  is  at  M  and  whose  circular  base  of  3"  diameter 
is  in  H  with  its  center  at  N.  The  cone  is  cut  by  a  plane  T(-  lO-]",  -  112^°,  +  150°). 
Find  the  H  and  V  projections  of  the  curve  of  intersection  of  the  plane  and  the 
cone,  the  true  size  of  this  section  when  swung  into  H  and  the  development  of  that 
portion  of  the  curve  between  the  plane  T  and  H.  (Develop  the  cone  in  the  right 
hand  portion  of  the  sheet,  placing  the  vertex  at  a  point  (-6^")  on  the  ground 
line.    State  problem  in  upper  central  part  of  sheet.) 

749.  [1]  The  line  M(- 13|",  +  2", +  4^")  N(- 13|",  +  2",  0")  is  the  axis  of  a 
right  circular  cone,  whose  vertex  is  at  M  and  whose  circular  base  of  3^"  diameter 
is  in  H  with  its  center  at  the  point  N.  The  cone  is  cut  by  the  plane 
T(-10^",  -  120°,  +  150°).  Find  the  H  and  V  projections  of  the  curve  of  inter- 
section of  the  cone  and  the  plane,  the  true  size  of  this  section  when  swung  into 
H,  and  the  development  of  that  part  of  the  cone  between  H  and  T.  (Develop 
the  cone  in  the  right  hand  portion  of  the  sheet,  placing  the  vertex  at  a  point 
(-  5^")  on  the  ground  line.     State  problem  in  the  upper  central  part  of  sheet. 

750.  [1]  P  at -or.  The  line  M(- 4^",  -  4i",  -  1^")  N(- 4^,  0",  -  If')  is 
the  axis  of  a  cone  whose  vertex  is  at  the  point  M  and  whose  base  is  an  ellipse  in 
V,  with  axes  2|"  and  1|"  respectively  in  length,  intersecting  at  the  point  N.  The 
major  axis  of  this  base  is  parallel  to  the  ground  line.  The  cone  is  cut  by  a  plane 
T(+ 1",  +  157|°,  -  150°).  Find  the  three  projections  of  the  cone,  and  the  curve 
of  intersection  of  the  cone  and  plane,  the  true  size  of  the  section  when  swung  into 
V,  and  the  development  of  that  part  of  the  cone  between  the  vertex  and  T.  (De- 
velop cone  in  upper  right  hand  portion  of  sheet,  stating  problem  in  lower  right 
hand  portion.) 

751.  [1]  P  at  -7i".  The  line  M(- 6",  -  3J",  -  2")  N(- 6",  0",  -  2")  is  the 
axis  of  a  right  circular  cone  whose  vertex  is  at  the  point  M  and  whose  circular 
base  of  3f  diameter  is  in  V  at  the  point  N.  The  cone  is  cut  by  a  plane 
T(- 3",+ 150°,  -  120°).  Find  the  three  projections  of  the  cone,  and  the  Hne  of 
intersection  of  the  cone  and  plane,  the  true  size  of  the  section  when  swung  into 
H,  and  the  development  of  that  part  of  the  cone  between  the  vertex,  the  plane 
T,  and  V.  (Develop  cone  in  upper  right  hand  portion  of  sheet,  stating  problem 
in  the  lower  right  hand  portion.) 

752.  [1]  P  at  -51".  The  line  M(- 2f",  -  2",  -  i")  N(- 2g",  -  2",  -  4")  is 
the  axis  of  a  right  circular  cone,  whose  vertex  is  at  the  point  M  and  whose  circular 
base  of  3|"  diameter  is  located  in  a  horizontal  plane  4"  below  H  with  its  center  at 
the  point  N.  The  cone  is  cut  by  the  plane  T(- 5f ,  + 60°,  -  45°).  Find  the 
three  projections  of  the  cone  and  the  line  of  intersection  of  the  cone  and  the 
plane,  the  true  size  of  the  section  when  swung  into  H,  and  the  development  of 
that  portion  of  the  cone  between  its  base  and  T.  (Develop  the  cone  in  the  left 
hand  portion  of  the  sheet,  placing  the  vertex  at  a  point  (-  6f )  on  the  ground 
line.     State  problem  in  upper  right  hand  part  of  sheet.) 

753.  [1]  P  at-9J/'.  The  line  M(- 2f ,  -  H",  -  J")  N(- 2|",  -  1^", -4")  is 
the  axis  of  a  right  circular  cone  whose  vertex  is  at  the  point  M  and  whose  circular 
base  of  2"  diameter  is  located  in  a  horizontal  plane  4"  below  H  with  its  center  at 


53 

the  point  N.  The  cone  is  cut  by  the  plane  T(- 6'',  +  45°,  -  45°).  Find  the  3 
projections  of  the  cone  and  curve  of  intersection  of  cone  and  plane  T,  the  true 
size  of  this  section  when  swung  into  H,  and  the  development  of  that  part  of  the 
cone  between  its  base  and  the  plane  T.  (Develop  cone  in  upper  right  hand  por- 
tion of  sheet.     State  problem  in  lower  right  hand  portion.) 

754.  [1]  P  at  -11".  The  line  M(- 3^'',  -  3",  -  If'')  N(- 2^',  0",  -  If")  is 
the  axis  of  a  right  circular  cone,  whose  vertex  is  at  the  point  M  and  whose  cir- 
cular base  of  2^"  diameter  is  in  V  with  its  center  at  the  point  N.  The  cone  is 
cut  by  the  plane  T(  oo',  -  3-|",  -  of").  Find  the  3  projections  of  the  cone  and  the 
curve  of  intersection  between  the  cone  and  the  plane,  the  true  size  of  the  curve 
of  intersection  when  swung  parallel  to  P  and  the  development  of  that  part  of  the 
cone  between  its  base  and  the  plane  T.  (Develop  the  cone  in  the  upper  right  hand 
portion  of  the  sheet.     State  the  problem  in  the  lower  right  hand  portion.) 

755.  [1]  P  at  -11".  The  line  M(-'2i",  -  If",  0")  N(- 2-^",  -  If",  -  4")  is 
the  axis  of  a  right  circular  cone,  whose  vertex  is  at  the  point  M  and  whose  circu- 
lar base  of  3"  diameter  is  located  in  a  horizontal  plane  4"  below  H  with  its  center 
at  the  point  N.  The  cone  is  cut  by  the  plane  T(  co,  -  f",  ^  2").  Find  the  3  pro- 
jections of  the  cone  and  curve  of  intersection  of  cone  and  plane,  the  true  size  of 
the  section  when  swung  into  the  plane  of  the  cone  base,  and  the  development  of 
that  part  of  the  cone  between  its  vertex,  the  plane  T  and  the  plane  of  its  base. 
(Develop  the  cone  in  the  upper  right  hand  portion  of  sheet;  state  problem  in 
lower  right  hand  portion.) 

756.  [1]  Profile  plane  at  -  11".  A  right  circular  cone  has  its  base  of  3"  diam- 
eter in  the  profile  plane,  and  its  axis  parallel  to  the  ground  line  through  the  ver- 
tex 0(-5",-2", -If").  It  is  cut  by  the  plane  T(- 5",  +  60°,  -  45°).  Find 
(1)  the  3  projections  of  the  curve  of  intersection  of  cone  and  plane,  (2)  the  de- 
velopment of  the  cone,  showing  this  intersection. 

757.  [1]  Profile  plane  at  -  10".  A  right  circular  cone  has  its  base  of  3"  diam- 
eter in  the  profile  plane,  and  its  axis  parallel  to  the  ground  line  through  the  ver- 
tex 0(-5". -2-i", -2^').  It  is  cut  by  the  plane  T(- 4^",  +  60°,  -  45°).  Find 
(1)  the  3  projections  of  the  curve  of  intersection  of  cone  and  plane,  (2)  the  de- 
velopment of  the  cone,  showing  this  intersection. 

7.— OBLIQUE  CONES  CUT  BY  SPHERES  AND  DEVELOPMENT. 

758.  [1]  The  point  0(-  10",  -  3|",  -  4^')  is  the  vertex  of  an  oblique  cone  and 
the  point  M(- 13f",  0",  -  2")  the  center  of  its  circular  base  of  2^"  diameter  in 
v.  The  cone  is  cut  by  a  sphere  of  6i"  diameter  whose  center  is  at  the  vertex  of 
the  cone.  Find  the  curve  of  intersection  and  develop  that  part  of  the  cone  be- 
tween the  sphere  and  V.  (Develop  cone  in  upper  right  hand  portion  of  sheet. 
State  problem  in  lower  right  hand  portion.) 

759.  [1]  The  point  0(-  14",  +  2f",  +  2f")  is  the  vertex  of  an  oblique  cone  and 
the  point  M(- lOf",  +  3",  0")  is  the  point  of  intersection  of  the  axes  of  its  ellip- 
tical base  in  H.  These  axes  are  respectively  4f"  and  3-|"  in  length,  the  major 
axis  being  parallel  to  the  ground  line.  The  cone  is  intersected  by  a  sphere  of  5" 
diameter  whose  center  is  at  the  vertex  of  the  cone.  Find  the  line  of  intersection, 
and  develop  that  part  of  the  cone  between  the  sphere  and  H.  (Develop  cone  in 
upper  right  hand  portion  of  sheet;  state  problem  in  lower  right  hand  portion.) 

760.  [1]  The  point  0(-  10",  -  4^",  -  i")  is  the  vertex  of  an  oblique  cone,  and 
the  point  M(- 13",  -  2",  -  4")  the  center  of  its  circular  base  of  3"  diameter,  in 
a  horizontal  plane  4"  below  H.  The  cone  is  cut  by  a  sphere  of  6"  diameter 
whose  center  is  at  the  vertex  of  the  cone.  Find  the  curve  of  intersection  and 
develop  that  part  of  the  cone  between  the  sphere  and  its  base.  (Develop  in  upper 
right  hand  portion  of  the  sheet.     State  problem  in  lower  right  hand  portion.) 


54 

8.— OBLIQUE  CONES  CUT  BY  PLANES  AND  DEVELOPMENT. 

761.  [1]  The  line  M(- 8^,  +  21'', +  4'')  0(- llf",  +  If",  0'')  is  the  axis  of 
an  obHque  cone,  whose  vertex  is  the  point  M  and  whose  circular  base  of  2^' 
diameter  is  in  H  and  its  center  at  the  point  O.  The  cone  is  cut  by  a  plane 
T(- 15",  -  55°,  +  20°).  Find  the  projections  of  the  curve  of  intersection,  the 
true  size  of  the  section,  and  the  development  of  the  cone  showing  the  curve  of 
intersection. 

763.  [1]  The  line  M(- 12",  +  2",  +  3")  0(- 12|",  +  2",-0")  is  the  axis  of  an 
oblique  cone  whose  vertex  is  the  point  M  and  whose  circular  base  of  3^"  diameter 
is  in  H  with  its  center  at  the  point  O.  The  cone  is  cut  by  a  plane 
T(- 10",  -  120°,  +  150°).  Find  the  projections  of  the  curve  of  intersection,  the 
true  size  of  this  section,  and  the  development  of  the  cone,  showing  the  curve  of 
intersection. 

763.  [1]  The  line  M(- 10",  -  4",  -  4")  0(- 12",  0",  -  H")  is  the  axis  of  an 
oblique  cone,  whose  vertex  is  the  point  M  and  whose  circular  base  of  3"  diameter 
is  in  V  with  its  center  at  the  point  O.  The  cone  is  cut  by  the  plane 
T(-  15",  +  30°,  -  60°).  Find  the  projections  of  the  curve  of  intersection,  the  true 
size  of  the  section,  and  the  development  of  the  cone  showing  the  curve  of  inter- 
section. 

764.  [1]  The  point  M(- 15^',  -  2^",  -  4|")  is  the  vertex  of  an  oblique  cone, 
and  the  axes  of  its  elliptical  base  in  V  intersect  at  the  point  0(-  12",  0", -1^"). 
These  axes  are  respectively  3"  and  2"  in  length,  the  former  being  parallel  to  the 
ground  Hne.  The  cone  is  cut  by  a  plane  T(- 15^",  +  22^°,  -  52^°).  Find  the 
projections  of  the  curve  of  intersection,  the  true  size  of  this  section,  and  the  devel- 
opment of  this  cone  showing  the  curve  of  intersection. 

765.  [1]  The  point  M(- 15f", -y',  - 1")  is  the  vertex  of  an  oblique  cone 
whose  circular  base  of  2|"  diameter  is  in  a  horizontal  plane  4^"  below  H  with  its 
center  at  the  point  0(- 11^",  -  3",  -  4^").  The  cone  is  cut  by  a  plane 
T(- 7|",  +  120°,  -  150°).  Find  the  projections  of  the  curve  of  intersection,  the 
true  size  of  this  section,  and  the  development  of  that  part  of  the  cone  between 
its  base  and  the  plane  T. 

766.  [1]  The  point  M(- 15f",  -  2|",  -  ^t")  is  the  vertex  of  an  oblique  cone, 
whose  base  is  a  circle  of  4"  diameter  located  in  a  plane  parallel  to  and  4|'"  below 
H,  with  its  center  at  the  point  0(- 11|",  -  2^",  -  4^").  The  cone  is  cut  by  a 
plane  T(- 8",  +  90°,  -  135°).  Find  the  projections  of  the  curve  of  intersection, 
the  true  size  and  shape  of  this  curve,  and  the  development  of  the  cone  between 
the  vertex,  the  plane  T  and  the  plane  of  the  base. 


9.— SPHERES  CUT  BY  PLANES. 

767.  [2]  A  sphere  of  4"  diameter  is  located  in  the  first  quadrant  with  its  cen- 
ter at  the  point  0(- 5|",  f  2^",  +  2|").  Find  the  curve  of  intersection  of  this 
sphere  with  a  plane  R^-  1",  -  135°,  +  135°). 

768.  [2]  A  sphere  of  3"  diameter  rests  upon  a  horizontal  plane  in  the  third 
quadrant  at  the  point  M(- 3|",  -  2",  -  3|").  Find  the  intersection  of  the  sphere 
with  a  plane  R(- 5", +  60°, -120°). 

769.  [2]  P  at  -  3y\  A  sphere  of  2|"  diameter  is  located  in  the  third  quadrant 
with  its  center  at  the  point  0(- If",  -  1|",  -  If").  Find  the  curve  of  intersec- 
tion of  this  sphere  and  a  plane  R(  oo,  -  3",  -  1^"). 


55 

10.— ELLIPSOIDS. 

Cut  by  planes;  true  size  of  curve  of  intersection. 

770.  [3]  An  ellipsoid  of  revolution  is  formed  in  the  third  quadrant  by  the  ro- 
tation about  its  vertical  major  axis  of  an  ellipse  whose  axes  are  respectively  3" 
and  2''.  The  ellipsoid  is  cut  by  a  plane  R  which  passes  through  the  middle  point 
of  its  vertical  axis  and  makes  an  angle  of  30°  therewith.  Find  (1)  the  curve  of 
intersection  between  the  surface  and  the  plane  R,  (2)  the  true  size  of  this  curve. 

771.  [2]  An  ellipsoid  of  revolution  is  formed  in  the  third  quadrant  by  the  rota- 
tion about  its  vertical  minor  axis  of  an  ellipse  whose  axes  are  3"  and  If  respec- 
tively. The  ellipsoid  is  cut  by  a  plane  R  which  intersects  the  vertical  axis  at  a 
point  ^"  below  its  highest  point  and  makes  an  angle  of  45°  therewith.  Find 
(1)  the  curve  of  intersection  between  the  surface  and  the  plane  R,  (2)  the  true 
size  of  this  curve. 

772.  [2]  An  ellipsoid  of  revolution  is  formed  in  the  third  quadrant  by  the 
rotation  about  its  vertical  major  axis  of  an  ellipse  whose  axes  are  4''  and  2^'' 
respectively.  The  ellipsoid  is  cut  by  the  plane  R  which  intersects  its  vertical  axis 
at  a  point  V'  below  its  highest  point  and  makes  angles  of  60°  and  30°  respectivelv 
with  H  and  V.  Find  (1)  the  curve  of  the  intersection  between  ellipsoid  and 
plane  R,  (2)  the  true  size  of  this  curve. 

773.  [2]  An  ellipsoid  of  revolution,  axes  6''  and  3''  respectively,  has  its  center 
at  the  point   0(- 4|",  +  2",  O'O,   and   its   long  axis   perpendicular   to   H.      Find 

(1)  the  intersection  of  the  ellipsoid  with  a  plane  R(-  2^",  -  90°,  +  135°),  (2)  the 
true  size  of  this  curve  of  intersection. 

11.— PARABOLOIDS  AND  HYPERBOLOIDS  OF  REVOLUTION. 

Intersection  by  planes;  true  size  of  curves  of  intersection;  lines  tangent  to 
intersection  curves. 

775.  [2]  The  line  A(-51^-2r,-D  B(- 1",  -  2^,  - 1")  is  the  directrix 
and  the  point  F(- 3^",  -  2:^",  -  1")  the  focus  of  a  parabola  which  is  the  genera- 
trix of  a  Paraboloid  of  Revolution,  formed  by  rotating  the  parabola  about  its  axis. 
Find  (1)  the  intersection  of  this  paraboloid  with  a  plane  R(- 5^",  +  67|°,  -  45°), 

(2)  the  true  size  of  this  curve  of  intersection,  (3)  a  line  tangent  to  the  curve  of 
intersection  at  its  lowest  point. 

776.  [2]  Find  (1)  the  curve  of  intersection  of  the  Paraboloid  in  Prob.  775 
with  a  plane  R,  parallel  to  V  and  f"  in  front  of  the  axis  of  the  paraboloid,  (2) 
the  true  shape  of  this  curve. 

777.  [2]  A  hyperbola,  in  a  plane  parallel  to  V,  has  its  foci  at  the  points 
F(-4i^-2i",-lf'')  and  G(- 4y^  -  2|",- 3i'')  and  its  vertices  at  the  points 
V(-  4:V',  -  2V',  -  2'')  and  W(-  4V',  -  UJ',  -  3").  This  hyperbola  spins  about  its 
vertical  transverse  axis  and  generates  a  Hyperboloid  of  Revolution.  Find  (1)  the 
curve  of  intersection  of  this  surface  with  the  plane  R(- 5:|'',  +  90°,  -  67^°),  (2) 
the  true  size  and  shape  of  this  curve,  (3)  a  line  PX  tangent  to  the  curve  of  inter- 
section at  the  highest  point  of  that  portion  of  the  curve  on  the  lower  nappe  of  the 
surface. 

778.  [1]  The  line  A(- 8^,  -  2'',  -  4")  B(- 81'',  -  2",  0'')  is  the  directrix  and 
the  point  F(- 7|",  -  2",  -  2'')  the  focus  of  a  parabola  which  is  the  generatrix  of 
a  Paraboloid  of  Revolution  with  AB  as  its  axis.  Find  (1)  the  intersection  of  this 
paraboloid  with  a  plane  R(- 7",  + 105°,  - 120°),  (2)  the  true  shape  of  that 
branch  of  the  curve  nearest  the  H  trace  of  R. 


56 

12.— TORUS. 

Intersection  by  plane;  tangent  line  to  curve  of  intersection  at  given  point. 

779.  [2]  A  Torus  is  formed  by  the  revolution  of  a  circle  of  1^"  diameter  about 
the  axis  A(-  4f",  +  2f ',  +  4'0  B(-  4-f ',  +  2-f' ,  0'').  The  circle  in  its  initial  posi- 
tion is  parallel  to  V  with  its  center  at  the  point  0(- 6^",  +  2|",  +  If ).  Find 
(1)  the  intersection  of  the  torus  with  the  plane  R(- If",  -  120°,  +  120°),  (2)  a 
line  tangent  to  the  curve  of  intersection  at  that  point  P  on  the  upper  portion  of 
said  surface  which  is  1"  in  front  of  V. 

780.  [2]  A  Torns  is  formed  by  the  revolution  of  a  circle  of  If  diameter  about 
the  axis  A(-3^^-2f  ,-3'0  B(- 31'',  -  2f ,  0").  The  circle  in  its  initial  po- 
sition is  parallel  to  V  with  its  center  at  the  point  0(- 4^",  -  2|'',  -  If").  Find 
(1)  the  intersection  of  the  torus  with  the  plane  R(- 7f ,  +  60°,  -  45°),  (2)  a  line 
tangent  to  the  curve  of  intersection  at  that  point  P  on  the  upper  portion  of  the 
surface  which  is  1^''  behind  V. 

781.  [1]  A  Torus  is  formed  by  the  revolution  of  a  circle  of  2"  diameter,  about 
the  axis  A(-  Q>V' ,  -  1^",  -  4'')  B(-  Q>^" ,  -  1^',  0")-  The  circle  in  its  initial  posi- 
tion is  parallel  to  V  with  its  center  at  the  point  0(- 7f'',  -  1|'',  -  2'').     Find 

(1)  the  intersection  of  the  torus  with  a  plane  R(- 13^",  +  17|°,  -  34J°),  (2)  the 
true  size  of  this  curve  of  intersection. 

782.  [1]  (G.  L.  par.  to  short  edge  of  sheet.)  A  Torus  is  formed  by  the  revo- 
lution of  a  circle  of  2]'''  diameter  about  the  axis  A(3f ,  -  5",  -  4|") 
B(- 3f ,  0'',  -  4^").  The  circle  in  its  initial  position  is  parallel  to  H  with  its 
center  at  the  point  0(- 6-J",  -  3'',  -  4^'').  Find  (1)  the  intersection  of  the  torus 
with  a  plane  R(- 10^'', -f- 45°,  -  60°),  (2)  a  line  tangent  to  the  curve  of  intersec- 
tion at  its  highest  point. 

13.— SQUARE  RING. 

Cut  by  plane ;  true  size  of  curve  of  intersection. 

783.  [1]  The  axis  of  a  Square  Ring  is  the  line  M(- 13'',  +  2^', +  |") 
N(- 13", +  2^", +  3").  The. generatrix  is  a  1^"  square  with  2  sides  parallel  to 
H,  whose  center  moves  in  a  horizontal  circle  of  3"  diameter  in  a  plane  If"  above 
H.     Find  (1)  the  intersection  of  the  ring  with  a  plane  T(- 8^",  -  90°,  +  157^°), 

(2)  the  true  size  of  this  intersection  when  swung  into  H. 

784.  [2]  The  axis  of  a  Square  Ring  is  the  line  M(- 5^",  +  2^", +  ^") 
N(- 5^",  + 2^",  +  3").  The  generatrix  is  a  1|"  square  with  2  sides  parallel  to 
H,  whose  center  moves  in  a  horizontal  circle  of  3"  diameter  in  a  plane  If"  above 
H.     Find  the  intersection  of  the  square  ring  and  a  plane  T(-^",  -  150°,  +  135°). 

785.  [2]  The  axis  of  a  Square  Ring  is  the  line  M(- 4^",  -  2f",  -  f ) 
N(- 4:J",  -  2f",  -  2-J-").  The  generatrix  is  a  If"  square  with  2  sides  parallel  to 
H,  whose  center  moves  in  a  circle  of  3Y'  diameter  in  a  plane  parallel  to  H  and 
If"  below  it.  Find  the  intersection  of  this  square  ring  with  the  plane 
T(-7",  +  45°,-45°). 

14._HYPERBOLIC  PARABOLOIDS. 

Assuming  elements ;  intersection  of  surface  by  planes ;  true  size  of  intersec- 
tion curves. 

794.  [2]  Thelines  M(-6",  +  f",  0")  N(- 4",  +  2",  +  2")  and  P(- 2",  +  1",  0") 
Q(- 1",  -  1:}",  +  If")  are  the  directrices  of  a  Hyperbolic  Paraboloid  whose  plane 
directer  is  H.  Find  the  intersection  of  this  surface  with  the  plane 
T(- 6^",  -  60°,  f  30°)  and  the  true  size  of  this  intersection. 


57 

795.  [2]  The  lines  M(-  6",  +  £",  0'')  N(-  4",  +  2",  +  2")  and  P(-  2'',  +  V,  0") 
0(- 1'',  -  1:^'',  +  If )  are  the  directrices  of  a  HyperboHc  Paraboloid  whose  plane 
directer  is  H.    Find  the  intersection  of  this  surface  with  V. 

796.  [1]  The  two  right  lines  M(- 131",  + 4|'',  +  3J-'')  N(- ll^'',  +  1'',  0'')  and 
P(-8i",  0",+3'')  Q(-5V',  +  5J:",  0")  are  the  directrices  of  a  Hyperbolic  Para- 
boloid, whose  plane  directer  is  D(- 13|",  -  150°,  +  120°).  Find  (1)  nine  ele- 
ments of  the  first  generation  through  points  on  the  directrix  MN  which  divide  it 
into  8  equal  parts,  (2)  the  projections  of  the  intersection  of  the  Hyperbolic  Para- 
boloid as  determined  by  these  elements  with  the  plane  T(- 11",  -  60°,  +  120°), 
(3)  the  true  size  of  this  curve  of  intersection  when  swung  into  H. 

797.  [1]  The  two  right  lines  M(- 12",  -  5^",  -  |")  N(- 6^:",  -  3",  -  4^")  and 
P(-llJ:",-l",-4")  0(-7f",-l",-l-f' )  are  the  directrices  of  a  Hyperbolic 
Paraboloid.  The  lines  MQ  and  PN  are  elements  of  the  first  generation.  Find 
(1)  thirteen  elements  of  the  first  generation  through  points  on  the  directrix  PQ 
which  divide  it  into  12  equal  parts,  (2)  the  projections  of  the  intersection  of  the 
Hyperbolic  Paraboloid  as  determined  by  these  elements  and  the  plane 
T(- 6", +  1574°,- 90°),  (3)  the  true  size  of  this  curve  of  intersection  when 
swung  into  H. 

798.  [1]  The  two  right  lines  M(- 12|",  -  5",  -  3")  N(- 12J-",  0",  -  3")  and 
P(-  8|",  0",  -  li")  Q(-  5i",  -  5",  -  4^-")  are  the  directrices  of  a  Hyperbolic  Par- 
aboloid whose  plane  directer  is  V.  Find  (1)  eleven  elements  of  the  first  genera- 
tion through  points  on  the  directrix  MN  which  divide  it  into  ten  equal  parts,  (2) 
the  projections  of  the  intersection  of  the  Hyperbolic  Paraboloid  as  determined  by 
these  elements,  with  the  plane  T(- 13^", +  45°,  -  45°),  (3)  the  true  size  of  this 
section  when  swung  into  V. 

799.  [1]  The  two  right  lines  M(- 13",  0",  -  1")  N(- 9",  -  3^",  -  3^")  and 
P(-  7",  0",  -  2")  Q(  -  5^",  -  3i",  -  2i")  are  the  directrices  of  a  Hyperbolic  Par- 
aboloid whose  plane  directer  is  V.  Find  (1)  nine  elements  of  the  first  generation 
through  points  on  the  directrix  MN  which  divide  it  into  8  equal  parts,  (2)  the 
projections  of  the  curve  of  intersection  of  the  Hyperbolic  Paraboloid  and  the  plane 
T(- 81", +  135°, -60°),  (3)  the  true  size  of  this  curve  of  intersection. 

800.  [2]  The  two  right  lines  A(- 6|",  -  I",  -  i")  B(- 5",  -  2",  -  4")  and 
C(-3",-J:",-3y')  D(-|",-4",-2")  are  directrices  of  a  Hyperbolic  Parabo- 
loid, of  which  BC  and  AD  are  elements.  Find  13  other  elements  of  the  same  gen- 
eration and  the  intersection  of  the  surface  with  the  plane  T(- 4^",  +  90°,  -  70°). 

801.  [2]  The  two  right  lines  A(- 61",  -  i",  -  i")  D(- i",  -  4",  -  2")  and 
B(-5", -2",-4")  C(-3", -Jr", -3^')  are  directrices  of  a  Hyperbolic  Parabo- 
loid of  which  AB  and  CD  are  elements.  Find  seventeen  other  elements  of  the 
same  generation  and  the  intersection  of  this  surface  with  the  plane 
T(-6", +  90°,-45°). 

802.  [2]  The  two  right  lines  A(- 6i",  -  V',  -  f)  B(- 5",  -  2", -4")  and 
C(-3", -J-/', -3^")  D(-^", -4", -2")  are  directrices  of  a  Hyperbolic  Parabo- 
loid of  which  AC  and  BD  are  elements.  Find  thirteen  other  elements  of  the  same 
generation  and  the  intersection  of  the  surface  with  a  plane  T(- 6",  +  90°,  -  60°). 

803.  [2]  The  two  right  lines  A(- 6^',  -  i",  -  i")  B(- 5", -4",  -  2")  and 
C(-3", -3|", -i")  D(-^", -2", -I")  are  directrices  of  a  Hyperbolic  Parabo- 
loid of  which  AC  and  BD  are  elements.  Find  thirteen  other  elements  of  this 
same  generation,  and  the  intersection  of  the  surface  thus  determined  with  a  plane 
T(-7i",  +  60°,-60°). 


58 

15.— HYPERBOLOIDS  OF  REVOLUTION  OF  ONE  NAPPE. 
Assuming  elements;  intersection  by  oblique  planes. 

804.  [1]  The  line  A(-  9'',  -  3^'',  -  2^)  B(-  9'',  0'^  -  2i'0  is  the  axis  of  a  Hy- 
perboloid  of  Revolution  of  one  Nappe,  located  in  the  third  quadrant  with  its  base 
in  V.  The  generatrix  in  its  initial  position  is  M(- 11",  0",  -  If") 
N(- 9'',  -  If",  -  1|").  Find  (1)  32  elements  of  one  generation  of  the  surface, 
which  pierce  V  at  equal  intervals  around  the  circumference  of  its  base,  (2)  the 
projections  and  true  size  of  the  curve  of  intersection  between  the  surface  and  a 
plane  T(- 14^",  +  25°,  -  45°). 

805.  [1]  The  line  A(- 11",  f  2f' ,  +  5")  B(- 11",  +  2|",  0")  is  the  axis  of  a 
Hyperboloid  of  Revolution  of  one  Nappe  which  stands  in  the  first  quadrant  with 
its  base  in  H.  The  generatrix  in  its  initial  position  is  M(- 13^',  +  2",  0") 
N(- 11",  +  2",  +  2|").  Find  (1)  32  elements  of  one  generation  of  the  surface 
which  pierce  H  at  equal  intervals  around  the  circumference  of  its  base,  (2)  the 
vertical  projection  of  the  meridian  section  parallel  to  V,  (3)  the  projections  and 
true  size  of  the  curve  of  intersecti  on  between  the  surface  and  a  plane 
T(-12|",-75°,  +  45°). 

806.  [2]  The  line  A(-  5",  -  2",  -  4")  B(-  5",  -  2",  0")  is  the  axis  of  a  Hyper- 
boloid of  Revolution  of  one  Nappe  located  in  the  third  quadrant  with  its  upper 
and  lower  bases  respectively  in  H  and  in  a  horizontal  plane  4"  below  H.  The 
generatrix  of  this  surface  is,  in  its  initial  position,  the  line  M(- 6^",  -  1^", -4") 
N(- 5",  -  1^",  -  2").  Find  (1)  32  elements  of  one  generation  of  the  surface, 
which  pierce  H  at  equal  intervals  around  the  circumference  of  its  base,  (2)  the 
projections  and  true  size  of  the  curve  of  intersection  of  the  surface  and  a  plane 
T(-10.85",  f  90°,-120°). 

807.  [2]  A  Hyperboloid  of  Revolution  of  one  Nappe  stands  in  the  1st  quad- 
rant with  its  base  of  4"  diameter  in  H  at  the  point  0(- 3^",  +  2i",  0").  The 
circle  of  the  gorge  of  2|"  diameter  is  in  a  horizontal  plane  1-J"  above  H,  with  its 
center  directly  above  O.  Find  32  elements  of  the  surface  and  the  intersection  of 
the  surface  thus  determined  with  the  plane  T(-  8",  -  54°,  +  23°). 

808.  [1]  (G.  L.  par.  to  short  edges  of  sheet.)  A  portion  of  a  Hyperboloid  of 
Revolution  of  one  Nappe  is  generated  by  the  revolution  of  the  line 
M(-6f",  hlf",  +  f")  N(-4y',-4f",  +  5i)  about  the  line  A(- 5|",  +  3",  +  |") 
B(- 5^",  +  3",  +  5^")  as  an  axis.  Assume  32  elements  of  the  surface  and  find 
the  intersection  of  the  surface  thus  determined  with  plane  T(-  10|",  -45°,  +  45°). 

809.  [1]  (G.  L.  par.  to  short  edges  of  sheet.)  Find  the  intersection  of  the 
surface  determined  in  the  above  problem  with  a  plane  R(- 10|",  -  60°,  +  39°). 

810.  [1]  (G.  L.  par.  to  short  edges  of  sheet.)  Find  the  intersection  of  the  sur- 
face determined  in  the  above  problem  with  a  plane  S(- 10",  -  52°,  +  38°). 

16.— HELICOIDS. 

Oblique  and  right.    Assume  elements;  intersections  with  oblique  planes, 
or  with  H  or  V ;  true  shape  of  curves  of  intersection. 

811.  [1]  The  line  A(- 12",  +  2",  +  4")  B(- 12",  +  2",  0")  is  the  axis  of  an 
oblique  Helicoid ;  the  generatrix  in  initial  position  is  M(- 13^",  +  2",  0") 
N(- 12", +  2",  +  H")  which  moves  in  such  a  way  that  its  horizontal  projection 
appears  to  rotate  counter-clockwise.  The  pitch  of  the  helix  which  is  generated 
by  the  point  M  is  2"  .  Find  (1)  sixteen  elements  of  the  surface  generated  during 
one  complete  movement  of  the  generatrix  about  the  axis,  (2)  the  intersection  of 
the  surface  thus  determined  with  the  H  plane  of  projection,  (3)  the  projections 


\ 


59 

and  true  size  of  the  curve  of  intersection  between  the  surface  thus  determined  and 
the  plane  T(- 9f',  -  90°,  +  150°). 

812.  [2]  The  line  A(- 4^,  -  If,  -  4'')  B(- 4^",  -  If",  0")  is  the  axis  of  an 
oblique  Helicoid,  whose  generatrix  in  initial  position  is  M(- 5-|",  -  If",  -  4") 
N(-4^", -If ,  -  lij")-  This  generatrix  moves  in  such  a  way  that  its  H  projec- 
tion appears  to  rotate  clockwise  and  the  point  M  generates  a  helix  whose  pitch 
is  2 1".  Find  (1)  sixteen  elements  of  the  surface  generated  during  1  complete 
movement  of  the  generatrix  about  the  axis,  (2)  the  intersection  of  the  surface 
thus  determined  with  a  horizontal  plane  4"  below  H,  (3)  the  projections  and  true 
size  of  the  curve  of  intersection  between  the  surface  thus  determined  and  the 
plane  T(- 8f ',  f  90°,  -  30°). 

813.  [1]  The  line  A(- lOV',  ~  4",  -  2")  B(- lOi",  0",  -  2")  is  the  axis  of  a 
Right  Helicoid  whose  generatrix  in  its  initial  position  is  determined  by  the  points 
M(-  9",  0'',  -  2")  and  N(-  10^'',  0",  -  2")-  This  generatrix  moves  in  such  a  way 
that  its  V  projection  appears  to  rotate  clockwise,  and  the  point  M  generates  a 
helix  whose  pitch  is  2".  Find  (1)  32  elements  of  the  surface,  generated  while 
the  generatrix  swings  about  the  axis  twice,  (2)  the  projections  of  the  curve  of 
intersection  between  the  surface  thus  determined  and  plane  T(-  7",  +  135°,  -  135°). 
Determine  and  mark  the  asymptotes  to  the  projections  of  the  intersection  curve 
in  both  H  and  V. 

814.  [1]  The  line  A(- 9'',  -  2-^,  0")  B(- 9'',  -  2^",  -  5'')  is  the  axis  of  a 
Right  Helicoid,  whose  generatrix  in  its  initial  position,  is  determined  by  the  points 
M (-  lir',  -  2|",  -  5'')  and  N (-  9",  -  2^',  -  5") •  This  generatrix  moves  in  such 
a  way  that  its  H  projection  appears  to  rotate  counter-clockwise,  while  the  point  M 
generates  a  helix  whose  pitch  is  2-^".  Find  (1)  sixty-four  elements  of  the  surface, 
generated  while  MN  swings  about  the  axis  twice,  (2)  the  curve  in  which  the  sur- 
face thus  determined  intersects  the  vertical  plane  of  projection.  Determine  and 
mark  the  asymptotes  to  this  curve. 

815.  [1]  (G.  L.  par.  to  short  edges  of  sheet.)  An  Oblique  Helicoid  is  given 
in  Prob.  610.  Find  its  intersection  with  H,  and  with  a  plane  T(-  10",-  67^°,+  30°). 

816.  [2]  Find  the  intersection  of  the  Oblique  Helicoid  of  Prob.  611,  (a)  with 
the  PI  plane  of  projection,  (b)  with  a  horizontal  plane  1^^"  above  H. 

817.  [2]  The  cast  iron  marking  post  in  Fig.  46  is  to  be  cut  off  by  a  plane  T, 
which  intersects  the  axis  1'  -  !{/'  below  the  top  of  the  post  and  makes  an  angle 
of  30°  with  said  axis.  Find  the  projections  of  the  curve  of  intersection  and  the 
true  size  and  shape  of  the  section  formed.     Scale,  1^-"  =  I'-O".. 

818.  [2]  One  side  of  the  marble  pillar  cap  of  Fig.  47  was  injured  and  in  order 
to  use  the  cap  as  an  ornament  in  another  place,  it  was  cut  off  by  the  plane  T, 
which  intersects  the  axis  at  a  point  2'  -  8"  above  the  bottom  plane  of  the  cap  and 
makes  an  angle  of  30°  with  the  axis.  With  scale,  f"  =  1'  -  0",  find  the  projections 
of  the  intersection,  and  the  true  size  of  the  section  formed. 

819.  [2]  Assume  a  convenient  scale  for  the  mortar  shown  in  Fig.  48  whose 
outer  surface  is  an  Hyperboloid  of  revolution  of  one  nappe  and  whose  interior 
is  partly  conical,  partly  spherical  as  indicated.  Find  the  projections  of  the  mortar, 
and  the  true  size  of  the  intersection  of  the  mortar  with  an  oblique  plane  cutting 
the  axis  at  a  convenient  angle. 

820.  [1]  G.  L.  par.  to  short  edge  of  sheet.  The  cast  iron  snubbing  post  shown 
in  Fig.  46  is  cut  by  a  plane  T(-  18'6'',  +  67|°,  -  60°).  Find  the  line  of  intersec- 
tion. The  axis  of  the  post  is"  the  line  A(- 11' 0",  -  8' 6",  0) 
B(-  11'  0",  -  8'  6",  -  16'  0")  and  its  base  stands  on  a  plane  which  is  14'  4^"  be- 
low H.    Draw  to  scale  of  2"  =  1'  0". 


6o 


SHORTEST   DISTANCE  BETWEEN   POINTS   ON   SURFACES. 


Prisms  and  Pyramids. 

825.  [1]  Find  the  shortest  distance  on  the  surface  from  the  point 
A(-  14|'',  y,  -  2|'0  on  the  front  to  the  point  B(-  13",  y,  -  i")  on  the  back  of  the 
right  prism  of  Prob.  725. 

826.  [1]  Find  the  projections  of  the  shortest  Hne  that  can  be  drawn  on  the 
surface  from  the  point  on  the  back  of  the  right  prism  of  Prob.  726  vertically  pro- 
jected at  A(- 14",  0'',  -  3i")  to  the  point  on  the  front  vertically  projected  at 
.B(-12r,0",-2"). 

827.  [1]  Find  the  shortest  distance  on  the  surface  from  the  point 
K(-3f",-t-li",  z)  to  the  point  L(- 4^",  +  2",  z)  lying  on  diflferent  faces  of  the 
oblique  prism  of  Prob.  727. 

828.  [1]  Find  the  projections  of  the  shortest  line  lying  on  the  surface  of  the 
right  pvramid  of  Prob.  730,  joining  the  points  on  its  surface  horizontally  projected 
at  A(-  3|",  +  2f' ,  0")  and  B  (-  4-|",  +  If',  0") . 

829.  [1]  Find  the  shortest  distance  along  the  surface  from  the  point 
C(-lli",-f",  z)  to  the  point  D(- 13^',  -  1^",  z)  on  the  surface  of  the  oblique 
pyramid  of  Prob.  731. 

830.  [1]  Find  the  projections  of  the  shortest  line  that  can  be  drawn  on  the 
surface  of  the  oblique  pyramid  of  Prob.  732  from  point  E(- 13^",  y,  -  1^")  to 
F(-  12f",  y,  -  If")  on  the  front  face. 

Cylinders  and  Cones. 

831.  [1]  Find  the  projections  of  the  shortest  line  that  can  be  drawn  on  the 
surface  from  the  point  A(-  15",  y,+  2")  on  the  front  to  the  point  B(-  13",  y,+  |") 
on  the  back  of  the  right  cylinder  of  Prob.  734. 

832.  [1]  Find  the  shortest  distance  on  the  surface  of  the  right  cylinder  of 
Prob.  736  from  the  point  C(-  4^",  y,  -  f")  on  the  back  of  the  surface  to  the  point 
D(-3|",  y,-H")  on  the  front  of  the  surface. 

833.  [1]  P  at  -  10".  Find  the  shortest  distance  on  the  surface  of  the  right 
cylinder  of  Prob.  738  from  the  point  E(- 3",  -  2§",  z)  on  the  bottom  to  point 
F(-  I",  -  li",  z)  on  top  of  the  cylinder. 

834.  [1]  Find  the  projections  of  the  shortest  line  that  can  be  drawn  on  the  sur- 
face of  the  oblique  cylinder  of  Prob.  739  from  the  point  A(- 12",  +  3",  z)  on  the 
top  to  the  point  B(-9|-",  +  2f",  z)  on  the  bottom  of  the  cylinder. 

835.  [1]  Find  the  shortest  distance  along  the  surface  from  the  point 
A(-10i",y, -2^")  on  the  front  to  point  B(- 12f' ,  y,  -  2")  on  the  back  of  the 
oblique  cylinder  of  Prob..  740. 

836.  [1]  P  at  -8^".  Find  the  projections  of  the  shortest  Hne  that  can  be 
drawn  on  the  surface  of  the  oblique  cylinder  of  Prob.  741  from  the  point 
C(-5i", -2|",  z)  on  the  top  to  point  D(- 3^",  -  If ',  z)  on  the  under  side  of 
the  surface. 

837.  [1]  Find  the  shortest  distance  from  point  A(-10f ,  + 1",  z)  on  the  top 
to  point  B(-12f',  +  3i",  z)  on  the  bottom  of  the  oblique  cylinder  of  Prob.  743. 

838.  [1]  Find  the  projections  of  the  shortest  line  that  can  be  drawn  on  the 
surface  of  the  right  cone  given  in  Prob.  747  from  C(-.10f ',  +  2f",  z)  to 
D(-12f',  +  4^z). 


6i 

839.  [1]  P  at  -of.  Find  the  projections  of  the  shortest  hne  that  can  be 
drawn  on  the  surface  of  the  right  cone  of  Prob.  752  from  point  E(-  3|",  -  1^",  z) 
to  point  F(-2'', -2V',  z). 

840.  [1]  P  at  -11".  Draw  the  projections  of  the  shortest  hne  that  can  be 
drawn  on  the  surface  of  the  right  cone  of  Prob.  756  from  point  A(-  I" ,  -  2%" ,  z) 
on  the  top  to  point  B(-  3^",  ~  If",  z)  on  the  under  side  of  the  surface. 

841.  [1]  Find  the  shortest  distance  on  the  surface  from  point  C(-  12^",  y,  -  3") 
on  the  front  to  point  D(- llf ,  y,  -  2|")  on  the  back  of  the  obHque  cone  of 
Prob.  758. 

842.  [1]  Find  the  shortest  distance  on  the  surface  of  the  obHque  cone  of  Prob. 
759  from  the  point  E(- lOf ,  +  2",  z)  on  the  top  to  point  F(- 13^",  +  3",  z)  on 
the  bottom  of  the  surface. 

Spheres. 

843.  [2]  Find  the  projections  of  the  shortest  hne  that  can  be  drawn  on  the 
surface  of  the  sphere  of  Prob.  767  from  point  A(- 6",  y,  +  1|")  on  the  front  to 
point  B(- 4|",  y,  +  31")  on  the  back  of  the  surface. 

844.  [2]  Find  the  shortest  distance  on  the  surface  of  the  sphere  of  Prob.  768 
from  point  C(-  4",  -  1^",  z)  on  the  top  to  point  D(-  2f",  -  2f',  z)  on  the  bottom 
of  the  surface. 

845.  [2]  P  at  -  3^".  Find  the  projections  of  the  shortest  Hne  that  can  be 
drawn  on  the  surface  of  the  sphere  of  Prob.  769  between  point  E(-  2|",  y,  -  If") 
on  the  front  and  point  F(-  1^",  y,  -  2^")  on  the  back  of  the  surface. 


II.     INTERSECTIONS  OF  TWO   SURFACES. 

1.— INTERSECTION  OF  TWO  CONES. 
Intersection  curves ;  line  tangent  to  said  curve,  or  development  of  one  cone. 

850.  [1]  (Take  ground  line  parallel  to  short  edge  of  sheet.)  The  axis  of  an 
oblique  cone  is  the  line  A(- 3^'',  +  f",  +  6")    B(- 6f",  +  3^'',  0'')-     The  base  of 

^this  cone  is  an  ellipse  in  H  with  center  at  B,  and  axes  3^''  and  2^"  respectively, 
the  latter  being  at  right  angles  to  the  H  projection  of  the  cone  axis.  The  line 
C(-6f",  +  2^'',  +  e3'^'')  D(-2|",  +  2f'',  0")  is  the  axis  of  another  oblique  cone, 
whose  circular  base  of  3f  diameter  is  in  H  with  its  center  at  D.  Find  (1)  the 
intersection  of  these  2  cones,  (2)  a  line  tangent  to  the  intersection  at  some  con- 
venient point. 

851.  [1]  The  line  A{-7V',- -y',0'')  B(- IH",  -  2",  -  4")  is  the  axis  of  the 
oblique  cone  whose  vertex  is  at  A  and  whose  circular  base  of  3"  diameter  is  in  a 
horizontal  plane  4"  below  H  with  its  center  at  B.  The  line  C(- 10",  0",  -  1^-") 
D(-  7|'',  -  3",  -  4")  is  the  axis  of  another  cone,  whose  vertex  is  at  C  and  whose 
circular  base  of  3"  diameter  is  in  the  same  horizontal  plane,  with  its  center  at  D. 
Find  (1)  the  intersection  of  these  2  cones,  (2)  a  line  tangent  to  the  curve  of  inter- 
section at  a  point  of  that  element  of  cone  AB  which  is  parallel  to  V. 

852.  [1]  A  right  circular  cone  stands  in  the  third  quadrant  on  a  horizontal 
plane  5"  below  H,  with  the  center  of  its  circular  base  (diameter  3")  at  the  point 
B(-ll'',-2",-5")-  Its  altitude  is  4V'.  The  line  C(- 12",  -  1^",  -  1|'0 
D(- 9|",  -  2^", -5")  is  the  axis  of  an  intersecting  oblique  cone,  whose  circular 
base  of  4"  diameter  is  in  the  same  horizontal  plane,  with  its  center  at  the  point  D. 
Find  (1)  the  intersection  of  these  two  cones,  (2)  the  development  of  the 
right  cone  showing  the  intersection. 

853.  [1]  A  right  circular  cone  of  32"  altitude  stands  on  H  with  the  center 
of  its  circular  base  of  2|"  diameter  at  the  point  B(- 13",  +  2",  0").  It  is  inter- 
sected by  an  oblique  cone  whose  vertex  is  at  the  point  C(- 15^",  +  2",  +  5")  and 
whose  circular  base  of  3"  diameter  is  in  H,  with  its  center  at  D(-  12|",  +  2",  0"). 
Find  (1)  the  intersection  of  the  two  cones,  (2)  the  development  of  the  right 
cone,  showing  this  intersection. 

854.  [1]  P  at  -6".  A  right  circular  cone  has  its  circular  base  of  3"  diameter 
in  the  profile  plane  and  its  axis  parallel  to  the  ground  line  through  the  vertex 
0(- 5",  -  2",  -  2").  Another  right  circular  cone  has  its  base  of  2V^  diameter 
these  two  cones. 

855.  [1]  The  line  A(- Hi",  +  4i",  + 1")  B(-7i",  0",  +  3i")  is  the  axis 
of  a  cone  whose  vertex  is  at  A  and  whose  base  is  a  3^'  circle  in  V  with  center  at  B. 
Line  C(-lli",  +  li",  !-4i")  D(-7i", -hlf",  0")  is  the  axis  of  a  second  cone 
whose  vertex  is  at  C  and  base  a  2V'  circle  in  H  with  center  at  D.  Find  the  curve 
of  intersection  of  the  two  cones. 


63 

2.— INTERSECTION  OF  TWO  CYLINDERS. 

Intersection  curves,  development  of  one  cylinder  or  line  tangent  to  curve  of 
intersection. 

850.  [1]  One  cylinder  cuts  clear  through  another.  The  line 
A(-10^  +  3^  +  2i'O  B(-7^0",  +  3i")  is  the  axis  of  a  cylinder  whose  circular 
base  of  2^'  diameter  is  in  V  with  its  center  at  the  point  B.  The  line 
A  M(- nf,  + 1^'',  0'')  is  the  axis  of  another  cylinder  whose  axis  intersects 
that  of  the  first  cylinder  at  A  and  whose  circular  base  of  IJ"  diameter  is  in  H 
with  its  center  at  the  point  M.  Find  (1)  the  intersection  between  these  two 
cyHnders,  (2)  the  development  of  the  larger  cylinder  showing  the  intersection. 

857.  [1]  The  line  A(- 11'',  -  4^',  -  1^")  B(- 14i",  -  2-]",  -  4f")  is  the  axis 
of  a  cylinder  whose  circular  base  of  3''  diameter  is  in  a  horizontal  plane 
4|''  below  II  with  its  center  at  the  point  B.  A  second  cylinder  has  the  line 
C(-14",-5",-f')  D(-ll",-lf',-4f')  for  its  axis  and  its  circular  base  of 
2|:''  diameter  is  in  the  same  horizontal  plane  as  the  base  of  the  first,  with  its  center 
at  the  point  D.  Find  (1)  the  intersection  of  the  two  cylinders,  (2)  the  develop- 
ment, showing  this  intersection,  of  the  larger  cylinder  between  the  base  and  a 
right  section  taken  above  the  intersection  of  the  two  cylinders. 

858.  [1]  A  line  AB  passing  through  the  point  A(- 12|",  +  3'',  + 3")  and 
making  angles  of  45°  with  H  and  30°  with  V  respectively,  is  the  axis  of  an 
oblique  cylinder  whose  base  of  1^"  diameter  is  in  H.  A  line  D(-  12|",  +  3",  +  2") 
E(-9f", +  1'',  0")  is  the  axis  of  another  cylinder  whose  circular  base  of  3|'' 
diameter  is  in  H  with  its  center  at  E.  Find  (1)  the  intersection  between  these 
cylinders,  (2)  the  development  of  the  smaller  cylinder. 

859.  [1]  The  line  A(- 8^'',  -  3",  -  4")  B(-ll",  0",  -  l^")  is  the  axis  of  an 
oblique  cylinder  whose  circular  base  of  2"  diameter  is  in  V  with  its  center  at  B. 
The  line  C(- 8;^,  -  2",  -  3")  D(-6",  0",-l")  is  the  axis  of  another  cylinder 
whose  circular  base  is  3V'  in  diameter,  and  is  in  V  with  its  center  at  D.  Find 
(1)  the  intersection  of  these  two  cylinders,  (2)  a  line  tangent  to  the  intersection 
at  a  point  on  the  highest  element  of  the  smaller  cylinder. 

860.  [l](Take  the  ground  line  parallel  to  short  edge  of  sheet.)  The  line 
A(- 5^",  +  }", +  5")  B(- 7V',  +  7",  0")  is  the  axis  of  a  cylinder  whose  circular 
base  of  2rr' diameter  is  in  H  with  its  center  at  B.  The  line  C(-  7^",  +  l^",  +  4f' ) 
D(- 31'',  +  5",  0'')  is  the  axis  of  another  cylinder  whose  base  is  an  ellipse  in  H 
with  its  center  at  D.  The  axes  of  this  ellipse  are  4|"  and  3^"  respectively  and  the 
minor  axis  is  at  right  angles  to  the  H  projection  of  the  cylinder  axis.  Find 
(1)  the  intersection  of  the  two  cylinders,  (2)  a  line  tangent  to  this  intersection 
at  some  convenient  point  P. 

861.  [11  Partially  cutting.  The  line  A (-13",- 4|",-li")  B(- 10^",  0",- l:?-") 
is  the  axis  of  a  cylinder  of  circular  right  section  of  2"  diameter  whose  base  is  in 
V.  The  line  Cf-ll", -3f",  -  If)  D(- 14f",  0",  -  2f")  is  the  axis  of  another 
cylinder  whose  circular  base  of  2"  diameter  is  in  V  with  its  center  at  D.  Find 
(1)  the  intersection  between  these  two  cylinders,  (2)  the  development  of  the  first 
mentioned  cylinder. 

862.  [1]  The  line  A(-  10",  -  U",  0")  B(-  14^',  -  1V\  -  4^')  is  the  axis  of  an 
oblique  cvlinder,  whose  circular  base  of  2"  diameter  is  in  a  horizontal  plane  iV' 
below  H  with  its  center  at  the  point  B.  The  line  C(- 13f",  +  2^",  0") 
D(- 11",  -  2:1",  -  4J")  is  the  axis  of  another  cylinder  whose  circular  base  of  4". 
diameter  is  in  the  same  horizontal  plane  with  its  center  at  D.  Find  (1)  the  inter- 
section of  these  2  cylinders,  (2)  the  development  of  the  smaller  cylinder. 


04 

8()3.  [2  I  A  right  circular  cylinder  in  the  3rd  quadrant  has  its  upper  base  of  3" 
diameter  in  H  with  its  center  at  the  point  A(- 6",  -  1^",  0")  and  its  axis  per- 
pendicular to  H.  Another  cylinder  has  a  circular  tapper  base  of  3"  diameter  in 
H  and  its  center  at  the  point  B(- 7",  -  3",  0").  Its  axis  is  in  a  plane  parallel 
to  V  and  the  vertical  projection  of  this  axis  makes  an  angle  of  45°  with  that  of 
the  axis  of  the  first  cylinder.  Find  (1)  the  intersection  of  these  2  cylinders, 
(2)  the  development  of  the  first  cylinder,  showing  the  intersection. 

864.  [2]  A  vertical  cylindrical  pipe,  diameter  3",  intersects  a  horizontal  cylin- 
drical pipe,  diameter  1-J".  Their  axes  are  1"  apart.  Find  the  projections  of  the 
curve  of  intersection,  and  develop  one  cylinder,  showing  this  intersection. 

8()5.  [2]  A  horizontal  cylindrical  pipe,  diameter  25",  is  joined  by  vertical  cyl- 
indrical pipe,  diameter  1  Y\  Their  axes  are  ^"  apart.  Find  the  projections  of  the 
intersection  curve,  and  develop  one  cylinder,  showing  this  intersection  curve. 

86().  [2]  Draw  the  projections  of  a  135°  elbow  for  a  2"  sheet  iron  pipe  and 
develop  one  branch. 

867.  [2]  Draw  the  projections  of  a  Y  intersection  for  a  5"  drain  pipe  and 
develop  one  branch  (Scale,  half  size). 

3._INTERSECTION  OF  CONE  AND  CYLINDER. 

Curves  of  intersection;  line  tangent  to  said  curves,  or  development  of  one 
surface. 

868.  [1]  (Take  ground  line  parallel  to  short  edge  of  sheet.)  The  point 
A(- 8^", +  ^",  +  6|")  is  the  vertex  of  an  oblique  cone  whose  elliptical  base  is  in 
H  with  its  center  at  the  point  B(-  3^",  +  5|",  0'').  The  axes  of  this  base  are  5" 
and  4"  respectively,  the  minor  axis  being  at  right  angles  to  the  H  projection  of 
the  axis  AB  of  the  cone.  The  line  C(-  5",  +  i",  +  5^")  D(-  6^",  +  6",  0")  is  the 
axis  of  an  oblique  cylinder  whose  base  of  2"  diameter  is  in  H  with  its  center  at  D. 
Find  (1)  the  intersection  between  the  cone  and  cylinder,  (2)  a  line  tangent  to  the 
intersection  curve  at  some  convenient  point. 

869.  [1]  The  point  A(- 10",  -  4^",  -  5^")  is  the  vertex  of  a  cone  which  ex- 
tends downward  into  the  third  auadrant  from  a  circular  base  of  5"  diameter  in 
H  with  its  center  at  the  point  B(-  5^',  -  2f' ,  0").  The  line  C(-  7",  -  5",  -  2^") 
D(- 12",  0",  -  2-i")  is  the  axis  of  a  cylinder  whose  circular  base  of  3"  diameter 
is  in  V  at  the  point  D.  Find  (1)  the  intersection  between  cone  and  cylinder, 
(2)  a  line  tangent  to  the  intersection  curve  at  some  convenient  point. 

870.  [1]  The  point  A(- 9",  -  5",  - 1")  is  the  vertex  of  an  oblique  cone  in  the 
third  quadrant,  whose  circular  base  of  3"  diameter  is  in  V  with  its  center  at 
B(-14",  0",-3f").  The  line  C(- 13^",  -  5|",  0")  D(- 10",  0",  -  3^")  is  the 
axis  of  an  oblique  cylinder  whose  circular  base  of  3"  diameter  is  in  V  with  its 
center  at  the  point  D.  Find  (1)  the  intersection  between  cone  and  cylinder, 
(2)  the  development  of  the  cone,  showing  the  intersection. 

871.  [1]  An  oblique  cone  has  its  vertex  at  the  point  A(-  9|",  -  3|",  -  5")  and 
its  circular  base  of  4"  diameter  in  V  with  its  center  at  the  point  B(-  13^,  0,  -  2^"). 
An  intersecting  oblique  cylinder  has  the  line  C(- 14^",  -  3",  -  4|") 
D(-  9^",  0",  -  2")  as  axis,  its  circular  base  of  2"  diameter  being  in  V  with  its 
center  at  the  point  D.  Find  (1)  the  intersection  between  cylinder  and  cone, 
(2)  the  development  of  the  cylinder,  showing  this  intersection. 

872.  [1]  The  point  A(- 11",  -  4^",  -  3^")  is  the  vertex  of  an  oblique  cone  in 
•the  3rd  quadrant  whose  circular  base  of  3"  diameter  is  in  V  with  its  center  at  the 
point  B(-13^",  0",-2").  The  line  C(- 13^',  -  3",  -  4^')  D(- 9|",  -  H",  0") 
is  the  axis  of  an  oblique  cylinder  whose  circular  base  of  2"  diameter  is  in  H  with 


6s 

its  center  at  the  point  D.    Find  (1)  the  intersection  of  cone  and  cyhnder,  (2)  the 
development  of  the  cyhnder  showing  the  intersection. 

873.  [11  An  obhque  cyhnder  in  the  3rd  quadrant  has  for  its  axis  the  hne 
A(-2", -V', -i'O  B(-6|'',-2'', -5^')  ;  its  circular  base  of  3"  diameter  is  lo- 
cated in  a  horizontal  plane  5''  below  H  with  its  center  at  the  point  B.  An  inter- 
secting oblique  cone  has  its  circular  base  of  2Y'  diameter  in  the  same  horizontal 
plane  with  center  at  a  point  D(- 3:|",  -  2f",  -  5'')  ;  its  vertex  is  at  the  point 
C(- 5|",  -  ^"..  -  f ).  Find  (1)  the  intersection  of  cone  and  cylinder,  (2)  the 
development  of  the  cylinder,  showing  this  intersection. 

874.  [1]  A  right  circular  cone  stands  in  the  third  quadrant  on  a  horizontal 
plane  5"  below  H.  Its  circular  base  of  3^"  diameter  has  its  center  at  the  point 
B(-  4'',  -  2^/',  -  5")  its  vertex  being  the  point  A(-  4",  -  2|",  -  V')-  An  oblique 
cylinder  has  the  line  C(- 6y',-V',- V')  D(- 3f',  -  3",  -  5")  "as  axis,  with  its 
circular  base  of  2"  diameter  in  the  same  horizontal  plane  as  the  cone,  with  its 
center  at  the  point  D.  Find  (1)  the  intersection  of  cone  and  cylinder,  (2)  the 
development  of  the  cone,  showing  this  intersection. 

875.  [1]  A  right  circular  cone  stands  in  the  third  quadrant  on  a  horizontal 
plane  5"  below  H.  Its  circular  base  of  3"  diameter  has  its  center  at  the  point 
B(-  4",  -  3V',  -  5"),  its  vertex  being  the  point  A(-  4",  -  3^',  -  ^").  An  oblique 
cylinder  has  for  its  axis  the  line  C(- 2'',  -  4^",  -  If)  DC"- 71",  -  H",  -  5")  its 
circular  base  of  2V'  diameter  being  in  the  horizontal  plane  above  mentioned  with 
its  center  at  the  point  D.  Find  (1)  the  intersection  of  concand  cylinder,  (2)  the 
development  of  the  cone,  showing  this  intersection. 

876.  [1]  An  oblique  cone  has  its  vertex  at  the  point  A(-  13|",  +  f',  +  of")  and 
its  circular  base  of  2|"  diameter  in  H  wnth  its  center  at  point  B(-  9§",  +  'iV',  0"). 
An  intersecting  oblique  cylinder  has  for  its  axis  the  line  C(- 11",  +  :|",  +  3^') 
D(- 13", +  3:5",0"),  with  its  circular  base  of  2-^-"  diameter  in  H  with  its  center 
at  the  point  D.  Find  (1)  the  intersection  between  the  cone  and  cylinder,  (2)  the 
development  of  the  cylinder  showing  this  intersection. 

877.  [1]  (Take  ground  line  parallel  to  shorter  edge  of  sheet  and  i"  above 
middle  of  sheet.)  The  line  A(-  6-J",  +  6|",  +  7")  B(-  3-1",  +  3,f',  0")  is' the  axis 
of  an  oblique  cylinder  whose  circular  base  of  2"  diameter  is  in  H  with  its  center 
at  the  point  B.  An  intersecting  oblique  cone  has  its  vertex  at  the  point 
C(- 2f",  +  7",  +  5^")  and  an  elliptical  base  in  H  with  its  center  at  the  point 
D(-  7f",  f  3",  0").  The  axes  of  this  ellipse  are  respectively  5"  and  4",  the  major 
axis  being  at  right  angles  to  the  FI  projection  of  the  cone  axis  CD.  Find  (1) 
the  intersection  of  cone  and  cylinder,  (2)  a  line  tangent  to  the  intersection  at  a 
point  P  which  is  3^  from  H,  on  that  part  of  the  curve  of  intersection  which  is 
visible  in  the  V  projection. 

878.  [1]  (Ground  line  parallel  to  short  edges  of  sheet,  and  one  inch  above  the 
middle  of  sheet.)  An  oblique  cone  has  its  vertex  at  the  point  A(-  8",  +  V\  +  G-J") 
and  the  center  of  its  elliptical  base  in  H  at  the  point  B(- 3",  +  5^",  0").  The 
axes  of  this  ellipse  are  respectively  5"  and  4",  the  minor  axis  being  at  right  angles 
to  the  H  projection  of  the  cone  axis  AB.  The  line  C(- 5",  + 11",  +  5|") 
D(- 6^",  +  7",  0")  is  the  axis  of  an  intersecting  oblique  cylinder  whose  circular 
base  of  2|"  diameter  is  in  H  with  its  center  at  the  point  D.  Find  (1)  the  inter- 
section of  cylinder  and  cone.  (2)  a  line  tangent  to  this  intersection  at  some  con- 
venient point. 

879.  [1]  (Ground  line  parallel  to  short  edges  of  sheet  and  1^"  above  middle 
of  sheet.)  An  oblique  cone  has  its  vertex  at  the  point  A(- 7^^",  -  6-|",  -  1")  and 
the  center  of  its  elliptical  base  in  V  at  the  point  B(-  2$",  0",  -  5").  The  axes  of 
the  ellipse  are  respectively  iV'  and  3f",  the  minor  axis  being  at  right  angles 


66  INTERSECTION  OF  CONE  AND  CYLINDER 

to  the  V  projection  of  the  cone  axis  AB.  The  Hne  C(- 4^'',  -  5", -^") 
D(- 6^'',  0'',  -  7'')  is  the  axis  of  an  intersecting  obHque  cyHnder  whose  circular 
base  of  2-Y'  diameter  is  in  V  with  its  center  at  the  point  D.  Find  (1)  the  inter- 
section of  the  cone  and  cyHnder,  (2)  a  line  tangent  to  this  intersection  at  some 
•convenient  point. 

880.  [1]  Line  A(- 14^",  +  3'',  +  3")  B(- 14^",  +  2'',  0")  is  the  axis  of  a  right 
circular  cone  vertex  at  A  and  center  of  3''  diameter  at  B.  Line 
C(-U-y',  +  3Y',  +  ^¥')  D(-11J",  0", +  11")  is  the  axis  of  a  right  circular  cyl- 
inder of  2"  diameter.  Find  (1)  the  base  of  cylinder  in  V,  (2)  the  intersection 
between  the  two  surfaces. 

88L  [1]  An  oblique  cone  has  its  vertex  at  the  point  A(-  9V',  -  SV',  -  5")  and 
its  circular  base  of  4"  diameter  in  V  with  its  center  at  point  B(-  13|'',  0",  -  2^"). 
An  intersecting  oblique  cylinder  has  the  line  C(- 14^'',  -  3",  -  4f") 
D(- 9^",  0",  -  If")  as  axis,  its  circular  base  of  2"  diameter  being  in  V  with  its 
center  at  the  point  D.  Find  (1)  the  intersection  between  cylinder  and  cone, 
(2)  the  development  of  the  cone,  showing  this  intersection. 

882.  [1]  The  point  A(- 11",  -  4|",  -  3^")  is  the  vertex  of  an  oblique  cone  in 
the  third  quadrant  whose  circular  base  of  3"  diameter  is  in  V  with  its  center  at 
point  B(-  131",  0",  -  2").  The  line  C(-  13|",  -  3^",  -  4|")  D(-  Of",  -  H",  0") 
is  the  axis  of  an  oblique  cylinder  whose  circular  base  of  2"  diameter  is  in  H  with 
its  center  at  the  point  D.  Find  (1)  the  intersection  between  cone  and  cylinder, 
(2)  the  development  of  the  cone  showing  this  intersection. 

883.  [1]  The  line  A(- 7-J", +  4i",  +  3|")  B(- 4",  0",  +  2|")  is  the  axis  of  an 
oblique  cone,  whose  vertex  is  at  A  and  whose  circular  base  of  3f "  diameter  is  in 
V  with  its  center  at  B.  An  intersecting  cylinder  has  for  its  axis  the  line 
C(-5",  +  4",  +  4")  D(-:)",  ^If",  0")  its  circular  base  of  3"  diameter  being  in 
H  with  its  center  at  the  point  D.  Find  (1)  the  intersection  of  cone  and  cylinder, 
(2)  a  line  tangent  to  this  intersection  at  a  convenient  point. 

884.  [1]  A  right  circular  cone  stands  in  the  third  quadrant  on  a  horizontal 
plane  5"  below  H,  with  the  center  of  its  circular  base  of  3|"  diameter  at  the  point 
B(-12", -2^", -5").  Its  altitude  is  4|".  An  obHque  intersecting  cylinder  has 
as  axis  the  line  C(- lo|",  -  f",  -  f")  D(- 11^",  -  3^",  -  5"),  and  has  its  circular 
base  of  3"  diameter  in  the  same  plane  as  the  cone  base  with  its  center  at  the 
point  D.  Find  (1)  the  curve  of  intersection  of  cone  and  cylinder,  (2)  the  devel- 
opment of  the  cone,  showing  this  intersection. 

885.  [1]  A  right  circular  cylinder  and  a  right  circular  cone  have  vertical  axes, 
and  bases  in  a  horizontal  plane  44"  below  H.  The  cone  base  is  of  4"  diameter 
with  center  at  the  point  B(-  13|",  -  2^",  -  4-i")  ;  the  cylinder  base  is  of  2"  diam- 
eter, with  center  at  the  point  D(- 14f",  -  1-|",  -  4^")  ;  the  cone  altitude  is  4", 
Find  (1)  the  intersection  of  cone  and  cylinder,  (2)  the  patterns  (developments) 
for  cone  and  cylinder,  showing  the  intersection  in  each  case. 

886.  [1]  A  right  circular  cylinder  and  a  right  circular  cone  have  horizontal 
axes,  and  bases  in  a  vertical  plane  4"  behind  V.  The  cone  base  is  of  4:|"  diame- 
ter, with  center  at  the  point  B(- 13",  -  4",  -  2|")  ;  the  cylinder  base  is  of  1^" 
diameter,  with  center  at  the  point  D(-  14^",  -  4",  -  2")  ;  the  cone  altitude  is  3|". 
Find  (1)  the  intersection  of  cone  and  cylinder,  (2)  the  patterns (  developments) 
for  cone  and  cvHnder,  showing  the  intersection  in  each  case. 


67 

4.— GENERAL  INTERSECTIONS  OF  TWO  SURFACES. 

Including  combinations  of  single-curved,  warped,  and  double-curved  sur- 
faces. 

887.  [1]  A  helical  convolute  is  formed  by  drawing  tangents  to  a  helix,  whose 
axis  is  the  line  M(- 13",  +  l^'',  +  4")  N(- 13",  +  l^",  0"),  and  which  is  gener- 
ated by  a  point  which  starts  from  a  point  0(- 14",  +  1^",  0")  in  H  and  moves 
in  such  a  way  that  its  horizontal  projection  appears  to  revolve  counter-clockwise. 
The  pitch  of  the  helix  is  4".  An  oblique  intersecting  cone  has  its  circular  base  of 
If"  diameter  in  V  with  its  center  at  a  point  B(- 10",  0",  +  ^")  and  its  vertex  at 
A(- 13",  +  3f",  +  If").  Find  (1)  the  intersection  of  cone  and  helical  convolute, 
(2)  the  development  of  the  helical  convolute,  showing  this  intersection. 

888.  !"1]  A  hemisphere  of  4"  diameter  stands  on  H  in  the  first  quadrant  with 
its  center  at  the  point  0(- 10^",  +  2^,  0").  It  is  cut  by  an  oblique  cylinder 
whose  axis  is  the  line  A(- 7^',  +  5^",  +  3^")  B(- 11",  +  2",  0")  and  whose  cir- 
cular base  of  2^"  diameter  is  in  H  with  its  center  at  B.  Find  (1)  the  intersection 
of  hemisphere  and  cylinder,  (2)  a  line  tangent  to  the  curve  of  intersection  at  some 
convenient  point. 

889.  [1]  A  hemisphere  of  4-^-"  diameter  stands  in  the  third  quadrant  on  a  hor- 
izontal plane  5"  below  H,  with  the  center  of  its  base  at  the  point 
0(- llg^",  -  2f",  -  5").  It  is  cut  by  an  oblique  cylinder,  whose  axis  is  the  line 
A(-  9",  -  2",  -  1")  B(-  13",  -  2",  -  5")  and  whose  circular  base  of  2|"  diameter 
is  in  the  above  horizontal  plane  with  its  center  at  the  point  B.  Find  (1)  the  inter- 
section between  hemisphere  and  cylinder,  (2)  the  development  of  cylinder,  show- 
ing the  intersection. 

890.  [1]  A  sphere  of  4"  diameter  is  in  the  third  quadrant  with  its  center  at 
the  point  0(- 9",  -  2^",  -  2^").  An  oblique  intersecting  cylinder  has  the  line 
A(-  6f",  -  2Y',  -  i")  B(-  11-i",  -  21",  -  5")  as  axis,  and  its  circular  base  of  2^" 
diameter  is  in  a  horizontal  plane  5"  below  H  wnth  its  center  at  the  point  B.  Find 
(1)  the  intersection  of  the  sphere  and  cylinder,  (2)  a  line  tangent  to  the  curve 
of  intersection  at  some  convenient  point. 

891.  [1]  A  sphere  of  4"  diameter  is  located  in  the  third  quadrant  with  center 
at  a  pomt  0(- 2f",  -  2^",  -  2^").  A  right  circular  cylinder  of  2"  diameter  has 
its  axis  parallel  to  the  ground  line  through  the  point  A(- 3",  -  2f",  -  3").  Find 
(1)  intersection  of  cylinder  and  sphere,  (2)  development  of  cylinder  showing  this 
intersection. 

892.  [1]  (G.  L.  par.  to  short  edge  of  sheet.)  An  ellipsoid  of  revolution  is 
generated  by  the  revolution  about  its  major  axis  of  an  ellipse  which  is  in  a  plane 
parallel  to  V,  with  its  center  at  the  point  0(- 5^",  +  2",  +  3")  and  with  axes  6" 
and  4"  respectively,  the  latter  being  parallel  to  H.  An  oblique  cone  has  its  ver- 
tex at  the  point  A(- 3^",  -  2^",  +  3|")  and  its  circular  base  of  4"  diameter  is  in 
H  with  center  at  B(-  7^",  +  2^",  0").  Find  the  intersection  of  cone  and  ellipsoid 
of  revolution. 

893.  [2]  A  sphere  of  5"  diameter  has  its  center  in  the  third  quadrant  at 
0(-2",'-2",-2^").  An  oblique  cone  has  the  vertex  A(- 2",  -  3^',  -  4i")  and 
its  circular  base  of  2-|"  diameter  is  in  V  with  center  at  the  point  B(-  5^",  0",  -  2"). 
Find  the  intersection  of  sphere  and  cone. 

894.  [2]  A  torus  with  axis  perpendicular  to  H  through  the  point 
0(-3|", +  4|",  0")  is  generated  by  a  circle  of  2"  diameter,  which  in  initial  posi- 
tion is  in  a  plane  parallel  to  V  with  its  center  at  the  point  M(-  5f",  +  4|",  +  1^"). 
A  right  circular  cone  of  4"  altitude,  stands  in  H  with  the  center  of  its  base  of 


68         .  gi!;neral  inteirsections  oi''  two  surface;s 

3|"  diameter  at  the  point  C(-4^",  +  2|^'',  0").     Find  the  projection  of  the  inter- 
section between  torus  and  cone. 

895.  [1]  A  torus  stands  on  H  in  the  first  quadrant.  Its  axis  is  the  hne 
M(-  9i'',  +  2^',  O'O  N(-  9^",  +  2-y',  +  3"),  and  its  generating  circle  of  If"  diam- 
eter in  initial  position  is  in  a  plane  parallel  to  V  with  center  at  the  point 
0(- 11|'',  +  2|^",  +  ^").  A  right  circular  cone  stands  upon  H,  with  the  center  of 
its  circular  base  of  4"  diameter  at  the  point  B(- 8|'',  +  S"*,  0").  Its  altitude  is 
3^''.     Find  the  intersection  of  torus  and  cone. 

896.  [1]  (G.  L.  par;  to  short  edge  of  sheet  and  f  above  middle  of  sheet.) 
A  torus  in  the  first  quadrant  has  the  line  M(-7",  f4",  0")  N(- 7'',  +  4", +  4") 
as  its  axis,  and  its  generating  circle  of  2|"  diameter  in  initial  position  is  in  a  plane 
parallel  to  H  with  its  center  at  the  point  0(- 9^'',  +  4'',  +  2").  An  oblique  inter- 
secting cone  has  its  circular  base  of  5"  diameter  in  V  with  center  at  point 
B(-3'',  +  6",  0''),  and  its  vertex  at  a  point  A(- 9'',  +  2'',  +  7").  Find  the  inter- 
section of  cone  and  torus. 

897.  [1]  A  torus  has  the  vertical  axis  A(- 12'',  -  4",  -  4")  B(- 12",  -  4'',  0") 
and  its  generating  circle  of  3|"  diameter  is  in  initial  position  in  a  plane  parallel 
to  V  with  its  center  at  0(- 14|",  -  4",  -  If").  Another  intersecting  torus  has 
the  horizontal  axis  C(- 7",  -  5|",  -  5^")  D(- 7",  0",  -  oi")  and  its  generating 
circle  of  4"  diameter  is  in  initial  position  in  a  horizontal  plane  with  its  center  at 
M(-  10",  -  2^",  -  B-Y').     Find  the  intersection  of  these  two  tori. 

898.  [1]  A  triangular  pyramid  stands  in  the  first  quadrant  with  its  vertex  at 
the  point  M(-  12",  +  2",  +  5")  and  the  three  vertices  of  its  triangular  base  at  the 
points  A(-13r,  +  §",0")  B(-12y',-4i",0")  and  C(- 9f",  +  If",  0").  It 
is  intersected  by  a  regular  triangular  prism  whose  lower  base  is  in  the  plane 
T(- 14^", -90°, +  120°)  with  its  center  at  a  point  1"  from  the  H  trace  of  the 
plane  T,  and  2"  from  V.  Each  side  of  the  base  is  2"  long,  that  side  nearest  V 
is  parallel  to  V,  and  the  altitude  of  the  prism  is  5^".  Find  (1)  the  intersection 
of  pyramid  and  prism,  (2)  the  development  of  each,  to  half  scale,  showing  this 
intersection. 

899.  [1]  A  regular  square  pyramid  of  4^"  altitude  has  the  axis 
M(-12i",-2i",-5")  N(-12y',-2i",-yO,  and  its  base  is  in  a  horizontal 
plane  5"  below  H,  with  the  line  A(- 12|",  -  f",  -  5")  B(- lOf",  -  If",  -  5")  as 
one  side.  A  right  circular  cylinder  of  1:|"  diameter  has  the  axis 
P(-14i", -2J", -3i")  Q(-10|",-2i",-lf").  Find  (1)  the  intersection  of 
pyramid  and  cylinder,  (2)  the  development  of  the  pyramid,  showing  the  inter- 
section. 

900.  [1]  The  lines  A(- 15",  0",- 5")  B(- 12",- 5",0")  and  C(- 8",- 5"- 5") 
D(- 3^",  0",  -  y')  are  the  directrices  of  a  hyperbolic  paraboloid  whose  plane 
directer  is  V.  It  is  intersected  by  an  oblique  cone  whose  vertex  is  the  pwint 
M(- 8y,  -  5f",  0")  and  whose  circular  base  of  4"  diameter  is  in  V  with  its  cen- 
ter at  the  point  N(-  10",  0",  -  3V').  Find  the  intersection  of  cone  and  hyperbolic 
paraboloid. 

901.  [1]  A  hyperbolic  paraboloid  has  the  lines  A(- 13f",  +  2^",  0") 
B(-10|",  +  3",-3")  and  C(- 13^",  +  1",  +  2^")  D(- lOJ", +  i",  0")  as  direc- 
trices and  H  as  plane  directer.  A  right  circular  cylinder  of  2"  diameter  lies  on 
H  with  the  line  M(- 14i", +  y',  +  1")  N(- 10",  +  3^",  +  1")  as  axis.  Find  (1) 
the  intersection  of  the  cylinder  and  hyperbolic  paraboloid,  (2)  the  development 
of  cylinder  showing  this  intersection. 

902.  [1]  The  right  line  directrices  of  a  certain  hyperbolic  paraboloid  are 
given  as  A(- 14^",  -  5f",  0")  B(- 11^",  +  T,  +  5'0  and  C(- 7^,  +  Sf",  +  5") 
D(- 2", -1- 1",  0")  and  H  as  plane  directer.     A  hyperboloid  of  revolution  of  one 


GENERAL   INTERSECTIONS   OF   TWO   SURFACES  69 

nappe  has  as  axis  the  line  E(- 9^',  +  2|",  0")  F(- 9^",  +  2|'',  +  5")  and  its  gen- 
eratrix in  initial  position  is  the  line  M(- 12^'',  +  f',  0")  N(- 9^'',  +  f",  +  3")- 
Find  (1)  the  meridian  curve  of  the  latter  which  is  parallel  to  V,  (2)  the  curve 
of  intersection  of  the  two  surfaces. 

903.  [1]  A  rig-ht  circular  cone  of  5"  altitude  stands  on  a  horizontal  plane  5:[" 
below  H.  with  the  center  of  its  circular  base  of  3^"  diameter  at  the  point 
Of- 13",  -  2^',  -  5:^").  An  oblique  helicoid  has  the  same  axis  as  the  cone,  its 
generatrix  in  initial  position  pierces  the  plane  of  the  cone  base  at  the  point 
M(-U-^",-2V',-H")  and  it  makes  an  angle  of  30°  with  the  axis.  Pitch  of 
helix  generated  by  point  M  =  t".  Find  (1)  the  intersection  of  cone  and  helicoid, 
(2)  the  development  of  the  cone,  showing  this  intersection. 

904.  [1]  A  right  circular  cone  stands  on  a  horizontal  plane  51''  below  H,  with 
the  center  of  its  circular  base  of  4"  diameter  at  the  point  0(- 12",  -  2f",  -  5|"). 
The  altitude  of  the  cone  is  6".  An  oblique  helicoid  has  the  same  axis  as  the  cone, 
its  generatrix  in  initial  position  is  the  line  M(- 12",  -  2^",  -  4^') 
N(-14y', -2J", -o-j")  and  its  pitch  is  2".  Find  (1)  the  intersection  of  cone 
and  helicoid,  carrying  the  generatrix  of  the  latter  surface  through  2  complete 
revolutions  about  the  axis,  (2)  the  development  of  the  cone  showing  the  inter- 
section. 

905.  [1]  An  oblique  helicoid  has  a  vertical  axis  which  pierces  H  at  the  point 
0(- 13|",  +  2",  0").  Its  pitch  is  4"  and  its  generatrix  in  initial  position  is  the 
line  A(-15", +  2",0")  B(- 13^',  +  2",  +  1|").  The  generation  is  such  that  in 
H  projection  the  line  generatrix  appears  to  rotate  counter-clockwise.  An  oblique 
cone  has  a  circular  base  of  3"  diameter  in  H  with  center  at  the  point 
N(-8|",  +  3|",  0")  and  vertex  at  M(- 13^", -1- 2",  +  2i").  Find  (1)  the  line  in 
which  the  helicoid  cuts  H,  (2)  the  intersection  of  helicoid  and  cone,  (3)  the  de- 
velopment of  cone  showing  the  intersection. 


III.     GENERAL  INTERSECTIONS. 


Including  three  surfaces. 

90G.  [1]  (Ground  line  parallel  to  shorter  edge  of  sheet.)  A  right  circular 
cylinder  of  3"  diameter  has  as  axis  the  line  A(-  7^",  +  5",  0")  B(-  3^",  0",  0")  ; 
a  sphere  of  4"  diameter  has  its  center  in  H  at  the  point  0(-  7^", +  4",  0")  ;  a  hy- 
perbolic paraboloid  whose  plane  directer  is  H,  has  directrices  M(- 7",  +  3",  +  3") 
N(-oi",  +  y',  0")  and  P(-3|^  +  2",  0'')  Q(- Sf,  0",  +  1|").  Find  (1)  the  in- 
tersection of  the  cylinder  with  the  sphere  and  hyperbolic  paraboloid,  considering 
only  that  part  of  each  surface  which  is  above  the  H  plane,  (2)  the  development 
of  the  semi-cylinder  between  the  intersecting  surfaces. 

907.  [1]  (Ground  line  parallel  to  shorter  edge  of  sheet.)  A  hexagonal  ring 
stands  on  H  in  the  first  quadrant  and  has  for  its  axis  the  line  M(-  6f",  +  3V',  +  3'') 
N(- 6f", -F  3|",  0").  Its  generating  hexagon  in  initial  position  stands  in  a  plane 
parallel  to  V,  with  its  base  in  H  and  its  center  at  the  point  P(-  8|",  +  3^",  + 1|")- 
A  regular  hexagonal  pyramid  of  4|"  altitude  stands  on  H  with  two  sides  of  its 
base  parallel  to  the  ground  line,  and  the  center  of  said  base  at  the  point 
0(- 8|",  +  3^",  0").  Sides  of  base  are  IJ".  An  oblique  cylinder  has  as  axis 
the  hne  A(- 2",  +  1^",  +  3^')  B(- 5^",  +  5",  0")  with  the  center  of  its  circular 
base  of  2^"  diameter  in  H  at  the  point  B.  Find  the  lines  of  intersection  of  the 
hexagonal  ring  with  the  pyramid  and  cylinder. 

908.  [1]  (Ground  line  parallel  to  shorter  edges  of  sheet.)  A  sphere,  an 
oblique  cylinder  and  an  oblique  cone  are  located  in  the  third  quadrant.  The 
sphere  is  of  5"  diameter  with  center  at  the  point  0(-  5|",  -  3",  -  3")  ;  the  cylin- 
der has  the  axis  A(- ,5J-",  -  3",  -  5fO  B(- 8^",  -  3",  0")  with  its  circular  upper 
base  of  2]''  diameter  in  H  at  the  point  B,  the  cone  has  its  vertex  at  the  center  of 
the  sphere,  and  its  elliptical  base,  axes  of  3"  and  3^"  respectively,  is  in  V  with 
center  at  the  point  M(- 3f ,  0'',  -  3")  and  with  its  major  axis  parallel  to  H. 
Find  the  intersections  of  the  sphere  with  cylinder  and  cone. 

909.  [1]  (Ground  line  parallel  to  shorter  edges  of  sheet.)  A  torus,  a  cylin- 
der and  a  cone  are  located  in  the  first  quadrant.  The  axis  of  the  torus  is  a  line 
perpendicular  to  H  through  the  point  M(- 5^",  +  4^",  0").  and  its  generating 
circle,  of  2^"  diameter,  in  initial  position  is  in  a  plane  parallel  to  \"  with  its  cen- 
ter at  the  point  X(- 7f",  +  4^", -i- 3|").  The  oblique  cylinder  has  as  axis  the 
line  A(-6V',  +  4V',  +  H")  B(- 9".  +  2",  0"),  the  center  of  its  circular  base,  of 
3''  diameter,  being  in  H  at  the  point  B.  The  vertex  of  the  cone  is  at  the  point 
C(- 5|", +  4|",  + 7")  and  the  center  of  its  circular  base  of  4''  diameter  is  in  H 
at  the  point  D(- 2y,  +  6|'',  0").  Find  the  intersections  of  the  torvis  with  cylin- 
der and  with  cone. 

910.  [1]  A  right  circular  cylinder  of  2^'  diameter  has  the  axis 
M(-13|",  0",  +  2'')  N(-6|'^  +  2'^  +  2");  a  sphere  of  5"  radius  has  its  center 
in  H  at  the  point  0(- -J'',  + 1'',  0")  ;  a  hyperbolic  paraboloid  has  H  as  plane 
director  and  as  directrices  the  lines  A(- 16",  0'',  +  4'0  B(- 13'',  +  5",  +  1")  and 
C(-6",  0",  +  4'0  D(-6'',  0'',  0'').  Find  (1)  the  intersections  of  the  cylinder 
with  the  sphere  and  hyperbolic  paraboloid,  (2)  the  development  of  that  part  of 
the  cylinder  between  the  other  two  surfaces,  said  development  to  be  made  on  a 
separate  sheet. 


GENERAL  APPLICATIONS 

BASED   ON 

INTERSECTIONS   AND   DEVELOPMENTS. 

915.  [2]  A  spiral  riveted  pipe  is  shown  in  Fig.  45.  Design  such  a  pipe, 
making  pitch  of  hehces  twice  the  pipe  diameter,  and  find  the  pattern  lay-out  for 
a  length  of  pipe  equal  to  four  times  the  diameter. 

916.  [1]  Fig.  49  shows  the  ventilator  connections  to  be  located  in  the  ridge  of 
a  factory  building.  Assume  a  convenient  scale  and  find  the  projections  of  the 
pipes  A,  B  and  D,  of  their  intersections,  and  the  patterns  (developments)  for  the 
same.    Also  the  pattern  for  a  cap  C,  to  overhang  3"  all  around. 

917.  [1]  A  sheet  metal  spire  is  shown  in  Fig.  55,  located  on  a  symmetrical 
gabled-  tower.  Assuming  reasonable  dimensions,  find  and  dimension  the  pattern 
for  the  spire. 

918.  [1]  A  stone  monument  is  shown  in  Fig.  59.  Assuming  a  convenient 
scale,  find  the  development  of  all  surfaces  except  the  sphere  and  to  scale  construct 
from  a  separate  sheet  a  model  for  the  monument. 

919.  [2]  A  sheet  metal  moulding  is  shown  intersecting  a  spherical  newel  post 
in  Fig.  GO.  Find  (1)  the  curves  of  intersection,  (2)  the  pattern  lay-out  for  the 
moulding. 

920.  [2]  A  symmetrical  triangular  galvanized  iron  panel  is  shown  in  Fig.  61. 
Find  the  projections  of  the  panel,  and  the  pattern  lay-out  for  one  side. 

921.  [1]  In  Fig.  62  is  shown  two  views  of  a  copper  cornice  ornament.  Con- 
struct the  projections  of  the  ornament,  and  find  and  dimension  the  complete 
pattern  lay-out. 

922.  [2]  A  galvanized  iron  pillar  base  is  shown  in  Fig.  63.  Find  the  projections 
of  the  base,  and  a  pattern  lay-out  for  the  intermediate  section. 

923.  [1]  In  Fig.  64  a  metal  gable  moulding  is  shown  at  its  intersection  with 
a  vertical  pilaster.  Assuming  convenient  scale  and  dimensions,  find  the  pro- 
jections of  the  intersection  curves,  and  pattern  for  a  short  length  of  both  moulding 
and  pilaster. 

924.  [1]  Design  a  cofifee-pot  with  conical  body,  spout  and  cover,  and  handle 
similar  to  that  shown  in  Fig.  65,  showing  its  projections  including  intersection 
curve  of  spout  and  body,  and  lay  out  patterns  for  cutting  the  metal  for  all  parts 
of  the  pot. 

925.  [1]  Scale,  f"  =  1'  -  0''.  An  iron  pouring-pot  has  a  cylindrical  body, 
conical  top  and  cylindrical  spout  as  indicated  in  Fig.  66.  Find  the  projections 
of  the  pot,  the  curves  of  intersection  and  the  developments  of  the  parts. 

926.  [2]  A  pipe  fitting  is  shown  in  Fig.  67.  Find  its  projections,  the  curves 
of  intersection,  and  develop  outer  and  inner  surface  of  one  branch  of  the  fitting. 

927.  [2]  Scale,  half  size.  A  copper  sink  drainer.  Fig.  68,  is  made  of  a  conical 
perforated  front  and  two  triangular  vertical  back  pieces  at  right  angles.  Con- 
struct the  three  projections  of  this  drainer  and  lay  out  the  pattern  to  cut  from  one 
piece  allowing  a  Y^  lap,  cut  ofif  at  the  correct  angle  at  each  end. 

928.  [1]  Scale,  1|"  =  I'-O".  Find  the  projections  and  pattern  development 
of  the  metal  reducer  in  Fig.  69  designed  to  connect  a  square  with  a  round  pipe. 
Dimension  in  full,  so  as  to  be  clear  to  the  one  who  lays  out  the  pattern. 


y2  GENERAL  APPLICATIONS 

929.  [2]  Scale,  14"  =  I'-O".  A  conical  offset  pipe  is  shown  in  Fig.  70,  connect- 
ing the  two  cylindrical  pipes.  Find  its  projections  and  lay  out  a  pattern  for  the 
same. 

930.  [2]  Find  the  projections  of  the  hopper  of  Fig.  71,  and  determine  all  the 
angles  which  would  be  necessary  in  construction.  Make  a  working  drawing  of 
one  side  of  the  hopper. 

931.  [1]  A  cylindrical  pipe  passes  through  a  factory  roof  in  Fig.  72.  Find  the 
projections  and  pattern  for  the  flange  as  shown,  assuming  the  flange  base  to  be 
a  square,  the  roof  to  be  inclined  at  45°,  and  the  center  line  of  the  pipe  to  pass 
through  the  center  of  the  square  flange  base. 

932.  [2]  Scale,  V  =  I'-O".  Show  three  views  of  the  bath  tub  of  Fig.  73,  and 
find  the  pattern  lay-out  for  the  same,  dimensioned  in  full  for  the  tinsmith. 

933.  [2]  A  tin  tea  pot  has  a  conical  spout  as  shown  in  Fig.  74.  Find  projec- 
tions and  pattern  for  this  spout. 

934.  [1]  A  tin  tea  pot  is  designed  with  conical  body  and  conical  spout  as  in 
Fig.  74.  Find  and  dimension  its  projections  and  complete  patterns.  Scale,  full 
size. 

935.  [2]  Construct  2  views  and  the  complete  pattern  for  the  steam  exhaust 
head  shown  in  Fig.  75. 

936.  [2]  Draw  two  views  and  lay  out  the  pattern  for  a  roof  flange  similar  to 
that  shown  in  Fig.  76,  assumed  to  be  a  portion  of  a  cone  of  revolution. 

937.  [1]  Scale,  1^'' =  I'-O".  Make  the  necessary  drawings  for  a  transition 
connection  between  square  and  round  pipes  similar  to  that  shown  in  Fig.  77. 
Lay  out  and  dimension  the  complete  pattern. 

938.  [1]  Design  a  metal  oil  can  as  shown  in  Fig.  78.  Show  necessary  views, 
and  lay  out  and  dimension  the  complete  pattern. 

939.  [2]  A  hexagonal  nut  has  a  conical  chamfer  as  shown  in  the  top  figure  of 
Fig.  79.  Construct  3  views,  assuming  a  =  2^",  d  =  \\" ,  and  the  chamfer  angle 
=  45°. 

940.  [2]  A  hexagonal  nut  has  a  spherical  top  as  shown  in  the  lower  figure  of 
Fig.  79.    Construct  3  views,  assuming  a  =  ^\" ,  d  =  \\" ,  r  =  1" ,   c  =  yV'. 

941.  [2]  An  upset  iron  rod  as  shown  in  Fig.  80,  is  turned  down  to  a  conical 
transition,  connecting  the  square  and  cylindrical  portions.  Construct  2  views  of 
such  a  rod,  finding  accurately  the  curves  of  intersection. 

942.  [1]  Design  a  metal  sugar  scoop  such  as  is  sketched  in  Fig.  81,  showing 
at  least  two  views  and  a  complete  pattern  for  the  same. 

943.  [1]  Construct  two  views  and  complete  pattern  lay-out  for  the  tin  meas- 
ure shown  in  Fig.  82. 

944.  [4]  Scale,  \"  =  I'-O".  The  Fig.  83  shows  a  hopper  used  on  a  rotating 
cylindrical  dryer.  The  hopper  top  is  an  18''  square,  the  sides  of  hopper  being 
faces  of  a  regular  pyramid  whose  vertex  is  in  the  center  line  of  the  cylinder  as 
shown,  b  =  2'-0''.  a  =  2'-6''.  Show  three  views,  find  intersection  of  hopper 
with  cylinder,  and  develop  pattern  for  one-half  the  hopper. 

945.  [2]  Two  cylindrical  air  shafts  intersect  as  shown  in  Fig.  84.  a  =  3'-0''. 
c  =  12''.  Center  line  of  small  shaft  is  9"  from  that  of  large.  Show  3  views  and 
develop  smaller  shaft.     Scale,  f "  =  I'-O". 

946.  [4]  Construct  to  convenient  scale,  two  views  of  the  stove  pipe  elbow  in 
Fig.  85.    Develop  pattern  for  one  half. 

947.  [2]  To  a  convenient  scale,  construct  two  views  of  the  furnace  elbow 
shown  in  Fig.  86,  and  lay  out  the  patterns  for  the  intermediate  section  and  one 
end  section. 


GENERAL  APPLICATIONS  73 

948.  [4]  Fig.  87  shows  a  square  furnace  pipe  elbow.  Construct  two  views, 
and  lay  out  a  pattern  for  the  complete  elbow. 

949.  [3]  Fig.  92  shows  a  water  tank  with  conical  boss.  Assuming  reasonable 
dimensions  and  scale,  show  2  views  of  the  tank  and  develop  the  boss  and  the 
tank. 

950.  [1]  A  locomotive  smoke  stack  is  dimensioned  in  Fig.  88.  To  scale, 
I"  =  I'-O'',  draw  two  views  of  the  same,  and  lay  out  sheets  for  the  several  sec- 
tions, dimensioning  ready  for  cutting. 

951.  [2]  A  pipe  connection  is  made  up  of  portions  of  a  torus  and  a  cylinder, 
flanged  as  shown  in  Fig.  89.  Find  two  views,  with  the  curve  of  intersection, 
and  develop  the  cylindrical  portion.     Scale,  1"  =  I'-O". 

952.  [2]  Design  a  tin  funnel  similar  to  that  shown  in  Fig.  90.  Show  2  views 
and  lay  out  complete  pattern,  dimensioning  in  full. 

953.  [4]  A  scale  scoop  is  formed  of  portions  of  2  equal  intersecting  cylinders 
as  in  Fig.  91.     Lay  out  and  dimension  a  complete  pattern. 

954.  [2]  A  portion  of  a  connecting  rod  is  shown  in  Fig.  93.  Assume  dimen- 
sions about  twice  the  size  of  the  blue  print  drawing,  and  find  three  projections  of 
the  curve  of  intersection  between  the  square  shaft  and  the  sphere  as  shown. 

955.  [1]  A  transition  connection  for  rectangular  to  round  pipe  is  shown  in 
Fig.  94,  with  round  pipe  development  and  transition  pattern.  Lay  out  and  di- 
mension the  complete  problem  to  scale,  3''  =  I'-O".  Develop  the  conical  surface 
BGE  by  Church's  method,  and  the  conical  surface  AGK  approximately  by  tri- 
angles. 

956.  [2]  Find  two  views  and  pattern  lay-out  for  the  furnace  partition  transi- 
tion pipe  shown  in  Fig.  95.     Scale,  1^"  =  I'-O''. 

957.  [1]  A  furnace  connection  is  to  be  constructed  as  shown  in  Fig.  96.  Find 
pattern  for  all  parts  shown,  assuming  reasonable  proportions  and  scale. 

958.  [2]  Design  a  metal  gable  moulding  and  wash  as  shown  in  Fig.  97  and 
find  the  pattern  lay-out  for  the  metal  cutter. 

959.  [2]  A  locomotive  slope  sheet  is  shown  in  Fig.  98.  Assume  reasonable 
dimensions  and  find  the  slope  sheet  pattern. 

960.  [1]  Design  two  gusset  plates  for  a  boiler  stack  such  as  indicated  in  Fig. 
99.  Show  three  views  of  the  plates  and  their  intersections  with  stack  and  boiler, 
then  find  the  development  of  one  of  these  gusset  plates. 


FIGURE  INDEX. 


The  following  is  a  list  of  blue  print  figures,  with  numbers  of  those  problems  which  are 
based  thereon. 


I-TGS.  PROBS. 

1-8  Introduction 

9  17,  18 

TO  48,  49,  92,  93>  94.  95 

II  74.  75.  82,  83,  84,  85 

12 
13 
14 
15 
16 
17 


18.... '..126  [4],  127  [4],  128 

295  [4] 

19 156  [4].  157  [4I.  158 

160  [4],  191  [4I.  208 

210  [4].  249  [4].  269 

401  [4].  402  [4],    426 

451  [2],  452  [2] 

20 t6i  [2],  211  [2],  267 

403  [2],  457  [2I,  464 

21 165  [8],  251 


[4],  294 


23 196  [2],  458 

24 434  [2],  454 

25- 
26. 
27. 


.163 

215 
435 


[2], 
[2], 

[2] 


28 166  [4], 

2r6  UL 

428  [2] 

29 192  [4I, 

270  [2], 

453  [2I. 

530  [2], 

.13T  [2], 


30. 


298  [2], 

31 129  [8], 

32 130  [8], 

433  [4] 

33 

34 

35 

36 

37 

38.. 

39 

40 

41 

42 

43 

44 

45 

46 


164  [2], 

324  [2], 

167  l4l. 
274  [4]. 

212  [4L 

271  [2], 

455  [2], 

531  [2], 

132  [2], 

431  [2] 

168  [8], 
162  [8], 

•517  [i]. 


•527 
193 
325 

194 
323 

213 
276 
456 
532 
169 

273 

272 

S18 
275 
277 


[I] 
[2] 
[2] 

[4l 
[4] 

[4l 
[2] 
[2] 
[2] 
[4] 

[8] 
[8] 

[I] 
[2], 
[2], 


•534  [i] 


.460  [4] 


.279 


.817 


[2] 
[2] 


.85 
.87 


..90 
..91 
[4]. 


,  159 

[4l 

,  209 

UJ 

.  322 

Ul 

.  427 

I2J 

,  268 

[2] 

.  521 

Ul 

.  522 

I2J 

•  523 

liJ 

,  524 

hJ 

,  525 

li] 

526 

[il 

1, 528 

III 

,  214 

l2| 

,  404 

I2J 

.  195 

[4] 

,  405 

I2J 

.  217 

[4l 

,  429 

[2\ 

.  529 

I2J 

,  197 

[2] 

.  430 

[4l 

,  432 

l4j 

,  5 19 

[I] 

296 

I2J 

297 

L2I 

533 

liJ 

535 

III 

536 

[2\ 

537 

[2\ 

,  542 

{2\ 

541 

\2\ 

538 

[I] 

539 

llJ 

.  540 

[2\ 

915 

|2| 

,  820 

llj 

FIGS. 


47- 

48. 

49. 
50. 

51- 
52. 
53- 
54- 

55- 


90. 
91. 
92. 

93- 

94. 

95- 
96. 
97- 
98. 
99. 


. . .  »I8 
...  819 
. . .  916 

614  [8] 

615  [8] 

616  [8] 

617  [8] 

618  [8] 
917 


56 619  [8] 


620 
621 
918 
919 
920 
921 
922 
923 
924 
92s 
926 

927 
928 

929 


57- 
58. 
59- 
60. 
61. 
62. 
63. 
64. 
65- 
66. 

67. 
68. 
69. 
70. 

71 543  [i],  930 

72 931 

73 932 

74 933  [2],  934 

75 935 

76 936 

77 937 

78 938 

79 939  [2].  940 

80. 
81. 
82. 
83. 
84. 
85. 
86. 

87. 


941 

942 

943 

944 

945 

946 

947 

948 

950 

951 

952 

953 

949 

954 

955 

544  [i],  956 

957 

958 

959 

960 


Fig.  I. 


Fig.  3. 


Fig.S. 


Fig.  2. 


Fig.  4. 


Fig. 6. 


Fig. 7. 


Fig.  8. 


OF  / 


■>p  the' 
VERSJTY 

OF 


Fig.  12. 


'4'.<" 


Fig.a 


'.   '^^  '' 


if?','V^ 


F/^./4. 


RK3/^*KTTna. 


F,-g./3. 


F,g.l6. 


Fig.  17. 


-* „0  -9/ 


..O^ir^  .,0-,0/     -*\ 


U      3f 


-*f*      3i" 


CAP  FOR   FA:7Vffy  /E/VT/LATOR  SH4FT. 


F/^.2/. 
Q^K  MISSION    LAMP    SHADE. 


rCEJSca/cr-S'.J 


F,cf.22. 


STEEL    TANK  FOR    AN    ELE)/ATOP    B007. 


P/afcs    to  be  bc^n^  afK/ 


THE 

iSlTY    ' 


8 -.3 

".f  - 

^ 

^ 

1 

:0       -    .-'« 


'9  ^/^/c/  /^^ 


'9  ^'^'^/d  P^J 


-.3-^   


V:       I 


1 

':£ 

H 

■*-^s>   -t*  -S 

1 
0 

•>*W          ^cv 

'^/  H  \*  ,.f/ 


IM 


m 


KSKI 


m 


\o-.oe 


W    t\l    <M 


.^V  ^ 


s  §  ^ 


^?^/v 


U--  ,0/       .  |«      ,31  — 


SY^—  ,0/  — 


■« *• 


1 
■ 

W^M 

ng.3/. 

MULE  TOiV/h/6    CAhJAL  BOAT 


"in" 


Fig.  3/ 


TU6-0r-lVAi?  ACROSS  RIVER. 


UNIVERSITY    FRESH-SOP^      SPRING    (JAMES. 


■N  /Ax : 


i 


iv^ 

^ 

Ch 

0. 

k^ 

k 

^ 

k'^ 

,6-ff  — H 


* '      OF    THE 

UNIVERSITY    ) 


■^'- 


•^ 

g^gr  — 

HfiSii^^^KiJ^'''  -  9 

— 

1 

Hj  34. 
A  COUNTRY    HOTEL     V 

= 

— 1 

■*©/->--«-  ,.o-p-  -A, 91 

^ 

''     THE 

.  .ISITY 


J 


J-—. 


I  /  Ml 

^ r  1 1 

^-iir 


-€^-H        h 


^1^ 

"  ch 

- 

:! 

-^''"  Fi^.  38. 

JACK  RAFTERS  B. 

SEE  SMALL  SKETCH. 


HOUSE  RAFTER  A. 


SEE  SMALL  SKETCH. 


\V^\ 


Fi^.  40. 


F/^.4/. 


I-BEAM    CONNECTION. 


BENT  PLATE  CHANNEL  CONNECTION 


i.  A 


m 


m 


m 


*'^     OF    THE 

UNIVERSIT' 

OF 


\i: 

HhI 

i      (!!■"•: 

-  '''"l 

ir 

1 

f-*--        /s^"     -*- 

F/^.  46. 
CAST  IRON  MARK/NG  POST. 


£(2)  Sca/e,  J"=^ /-oj 


-    ,1- 

■HUi^H 

\ 

^^wSaHB 

/®k  .7',™^ 

'I 

i_ 

^' 

v^^'fV 

'V/f#\^>-^ 

'i      ■' 

vo 

^ 

^^^^BumlUHHuHiiMBRH 

Pl^lfe 

MARBLE   PILLAR  CAR 

[fa).Sca/e,i''=/'-0''} 

F,g.48. 
MORTAR. 


f 

S      ; 

\  ; 
V 

\ 

■■*-           f  -    /7'        *-H 

/ 
/ 

F/-^.49. 
i^ENTILATOR    CONA/FCTIONS. 


Of    THE 

UNIVERSITY   ) 

OF  / 


m 


^M 


/  OF    THE 

f    UNIVERSITY 
V  ^^ 


'r^  „^T^  — 


\ 

-< 

;■' 

^ 

i^ 

) 

1 

1 

1          1 
1          1 
1         1 

V                               / 

'/ 

.5^   ^ 
1 

"^  1 

n 

< 

17 

■^                 .,9               ^ 

^^    THE  \ 

UMIVERSfTY    I 

Of  J 


1,?^  fi 


■I 


^  ^  ./.  ^  Jp/ 


i 

\ 

\ 

C^ 

\ 

\ 

\ 

t 

■• 

i 

-               ..0-3                .- 

..S/            — 

\  O    0   0    0     Q     0 
n    ^    0    . 


EH 

! 
1 

v^""' '  ^ 

f 

V. 

.^-^-"^'''^^^ 

i^;.^ 

- 

^^^^ 

1 -„s-je     *, 

r^  ^ 


p  -^ 


u 


-'^    THE 

OF 


\L^.^ 


SQUARE     HOPPER 


ROTARY  CYLINDRICAL  DRYER. 


INTERSECTING  AIR  SHAFTS. 


C/rcu/c»-    sac^ion. 


E/<^.3S. 


Fi^ae. 

FURNACE  AND  STO^E-PIRE    ELBOl^S. 


Fig.  87 


Ff'g.ae. 


LOCOMOTIVE  SMOKE   STACK 


I 


Fig.  89. 


FLANGED   PIPE   CONNECTION. 


r/ci.  92. 
WATER  TANK  WfTH  COMCAL  BOSS. 


SCALE  SCOOR 


V 

\mam. 

— \_ 

--A 

1 

J 

^> 

^  "T 

eA 

1    i 

^ 

Jl 

Fia.  93. 


CONNECTING-ROD    END. 


TRANSITION   CONNECTION. 


FftOM    ReCTANSCfLA/t      TO    ftOUND     PIPe . 


OF   THE 

UNIVERSITY 

Of 


— ^"- 

u 

sy 

\ 

\ 

1 

\ 

1     '— 

^^ 

A 

ry 

1 

I 

1 
1 

r 
1 

/ 

-» —  s-.i  -*- 

iC     ^ 


I 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 
BERKELEY 

Return  to  desk  from  which  borrowed. 
This  book  is  DUE  on  the  last  date  stamped  below. 


<^^^ 


s^"^ 


f^<^ 


^^f^' 


RECD  i-^ 
NOV    5  19S7 

JUN  12  1958     R 


asep 


b^VV- 


6Noy'57WW 


DEC0  3  199Z 


LD  21-100m-9.'47(A57028l6)476 


/ 


a4GG9 


F5- 


YE  02300 


U.C.  BERKELEY  LIBRARIES 


lllllllllllllll  II 


